{"id":164,"date":"2019-08-20T17:03:34","date_gmt":"2019-08-20T21:03:34","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/vectors\/"},"modified":"2022-06-01T10:39:35","modified_gmt":"2022-06-01T14:39:35","slug":"vectors","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/vectors\/","title":{"raw":"Vectors","rendered":"Vectors"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section you will:\n<ul>\n \t<li>View vectors geometrically.<\/li>\n \t<li>Find magnitude and direction.<\/li>\n \t<li>Perform vector addition and scalar multiplication.<\/li>\n \t<li>Find the component form of a vector.<\/li>\n \t<li>Find the unit vector in the direction of\u2009[latex]v[\/latex].<\/li>\n \t<li>Perform operations with vectors in terms of\u2009[latex]i[\/latex]\u2009and\u2009[latex]j[\/latex].<\/li>\n \t<li>Find the dot product of two vectors.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135472124\">An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140\u00b0. A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_08_001\">(Figure)<\/a>. What are the ground speed and actual bearing of the plane?<\/p>\n\n<div id=\"Figure_08_08_001\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152829\/CNX_Precalc_Figure_08_08_001.jpg\" alt=\"Image of a plan flying SE at 140 degrees and the north wind blowing\" width=\"487\" height=\"462\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\nGround speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane\u2019s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let\u2019s examine the basics of vectors.\n<div id=\"fs-id1165137737683\" class=\"bc-section section\">\n<h3>A Geometric View of Vectors<\/h3>\n<p id=\"fs-id1165135203508\">A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:<\/p>\n\n<ul>\n \t<li>Lower case, boldfaced type, with or without an arrow on top such as <strong>[latex]v,\\,\\,u,\\,\\,w,\\,\\,\\stackrel{\\to }{v},\\,\\,\\stackrel{\\to }{u},\\,\\stackrel{\\to }{w}.[\/latex]<\/strong><\/li>\n \t<li>Given initial point[latex]\\,P\\,[\/latex]and terminal point[latex]\\,Q,\\,[\/latex]a vector can be represented as[latex]\\,\\stackrel{\\to }{PQ}\\,.\\,\\,[\/latex]The arrowhead on top is what indicates that it is not just a line, but a directed line segment.<\/li>\n \t<li>Given an initial point of[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminal point[latex]\\,\\left(a,b\\right),\\,[\/latex]a vector may be represented as[latex]\u2329a,b\u232a.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137762492\">This last symbol [latex]\u2329a,b\u232a[\/latex] has special significance. It is called the standard position. The <span class=\"no-emphasis\">position vector<\/span> has an initial point [latex]\\left(0,0\\right)\\,[\/latex]and a terminal point[latex]\u2329a,b\u232a.[\/latex]To change any vector into the position vector, we think about the change in the <em>x<\/em>-coordinates and the change in the <em>y<\/em>-coordinates. Thus, if the initial point of a vector[latex]\\,\\stackrel{\\to }{CD}\\,[\/latex]is[latex]\\,C\\left({x}_{1},{y}_{1}\\right)\\,[\/latex]and the terminal point is[latex]\\,D\\left({x}_{2},{y}_{2}\\right),\\,[\/latex]then the position vector is found by calculating<\/p>\n\n<div id=\"fs-id1165135427119\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\stackrel{\\to }{AB}\\,=\\,\u2329{x}_{2}-{x}_{1},{y}_{2}-{y}_{1}\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\u2329a,b\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135160008\">In <a class=\"autogenerated-content\" href=\"#Figure_08_08_003\">(Figure)<\/a>, we see the original vector[latex]\\,\\stackrel{\\to }{CD}\\,[\/latex]and the position vector[latex]\\,\\stackrel{\\to }{AB}.[\/latex]<\/p>\n\n<div id=\"Figure_08_08_003\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152837\/CNX_Precalc_Figure_08_08_003.jpg\" alt=\"Plot of the original vector CD in blue and the position vector AB in orange extending from the origin.\" width=\"487\" height=\"290\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165135452050\">\n<h3>Properties of Vectors<\/h3>\n<p id=\"eip-id1165137910834\">A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at[latex]\\,\\left(0,0\\right)\\,[\/latex]and is identified by its terminal point[latex]\u2329a,b\u232a.[\/latex]<\/p>\n\n<\/div>\n<div id=\"Example_08_08_01\" class=\"textbox examples\">\n<div id=\"fs-id1165134568991\">\n<div id=\"fs-id1165134568994\">\n<h3>Find the Position Vector<\/h3>\n<p id=\"fs-id1165137736871\">Consider the vector whose initial point is[latex]\\,P\\left(2,3\\right)\\,[\/latex]and terminal point is[latex]\\,Q\\left(6,4\\right).\\,[\/latex]Find the position vector.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137655654\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137655654\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137655654\"]The position vector is found by subtracting one <em>x<\/em>-coordinate from the other <em>x<\/em>-coordinate, and one <em>y<\/em>-coordinate from the other <em>y<\/em>-coordinate. Thus\n<div id=\"fs-id1165137893431\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u23296-2,4-3\u232a\\hfill \\\\ \\,\\,\\,=\u23294,1\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137817747\">The position vector begins at[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminates at[latex]\\,\\left(4,1\\right).\\,[\/latex]The graphs of both vectors are shown in <a class=\"autogenerated-content\" href=\"#Figure_08_08_022\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_08_022\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152840\/CNX_Precalc_Figure_08_08_022.jpg\" alt=\"Plot of the original vector in blue and the position vector in orange extending from the origin.\" width=\"487\" height=\"349\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137532813\">We see that the position vector is[latex]\u23294,1\u232a.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137766874\">\n<div>\n<h3>Drawing a Vector with the Given Criteria and Its Equivalent Position Vector<\/h3>\n<p id=\"fs-id1165135390738\">Find the position vector given that vector<strong>[latex]\\,v\\,[\/latex]<\/strong>has an initial point at [latex]\\,\\left(-3,2\\right)\\,[\/latex]and a terminal point at[latex]\\,\\left(4,5\\right),\\,[\/latex]then graph both vectors in the same plane.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137827504\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137827504\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137827504\"]\n<p id=\"fs-id1165134041193\">The position vector is found using the following calculation:<\/p>\n\n<div id=\"fs-id1165134041196\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u23294-\\left(-3\\right),5-2\u232a\\hfill \\\\ \\text{ }=\u23297,3\u232a\\hfill \\end{array}[\/latex]<\/div>\nThus, the position vector begins at[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminates at[latex]\\,\\left(7,3\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_08_004\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152847\/CNX_Precalc_Figure_08_08_004n.jpg\" alt=\"Plot of the two given vectors their same position vector.\" width=\"487\" height=\"328\"> <strong>Figure 4.<\/strong>[\/caption]\n<p id=\"fs-id1165135347493\">[\/hidden-answer]<span id=\"fs-id1165135538542\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135169334\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_08_01\">\n<div id=\"fs-id1165137596741\">\n<p id=\"fs-id1165137596742\">Draw a vector<strong>[latex]\\,v\\,[\/latex]<\/strong>that connects from the origin to the point[latex]\\,\\left(3,5\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137834146\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137834146\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137834146\"]<span id=\"fs-id1165137656713\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152849\/CNX_Precalc_Figure_08_08_006.jpg\" alt=\"A vector from the origin to (3,5) - a line with an arrow at the (3,5) endpoint.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137551413\" class=\"bc-section section\">\n<h3>Finding Magnitude and Direction<\/h3>\n<p id=\"fs-id1165134381525\">To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.<\/p>\n\n<div id=\"fs-id1165135209432\">\n<h3>Magnitude and Direction of a Vector<\/h3>\n<p id=\"fs-id1165137422586\">Given a position vector<strong>[latex]\\,v[\/latex]<\/strong>[latex]=\u2329a,b\u232a,[\/latex]the magnitude is found by[latex]|v|=\\sqrt{{a}^{2}+{b}^{2}}.[\/latex]The direction is equal to the angle formed with the <em>x<\/em>-axis, or with the <em>y<\/em>-axis, depending on the application. For a position vector, the direction is found by[latex]\\,\\mathrm{tan}\\,\\theta =\\left(\\frac{b}{a}\\right)\u21d2\\theta ={\\mathrm{tan}}^{-1}\\left(\\frac{b}{a}\\right),\\,[\/latex]as illustrated in <a class=\"autogenerated-content\" href=\"#Figure_08_08_017\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_08_017\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152902\/CNX_Precalc_Figure_08_08_017new.jpg\" alt=\"Standard plot of a position vector (a,b) with magnitude |v| extending into Q1 at theta degrees.\" width=\"487\" height=\"216\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137730380\">Two vectors <strong><em>v<\/em><\/strong> and <strong><em>u<\/em><\/strong> are considered equal if they have the same magnitude and the same direction. Additionally, if both vectors have the same position vector, they are equal.<\/p>\n\n<\/div>\n<div id=\"Example_08_08_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137619099\">\n<div>\n<h3>Finding the Magnitude and Direction of a Vector<\/h3>\n<p id=\"fs-id1165137838002\">Find the magnitude and direction of the vector with initial point[latex]\\,P\\left(-8,1\\right)\\,[\/latex]and terminal point[latex]\\,Q\\left(-2,-5\\right).[\/latex]Draw the vector.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135413661\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135413661\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135413661\"]\n<p id=\"fs-id1165135413663\">First, find the <span class=\"no-emphasis\">position vector<\/span>.<\/p>\n\n<div id=\"fs-id1165137433330\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}u=\u2329-2,-\\left(-8\\right),-5-1\u232a\\hfill \\\\ \\text{ }=\u23296,-6\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134466815\">We use the Pythagorean Theorem to find the magnitude.<\/p>\n\n<div id=\"fs-id1165134466819\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|u|=\\sqrt{{\\left(6\\right)}^{2}+{\\left(-6\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{72}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=6\\sqrt{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137428177\">The direction is given as<\/p>\n\n<div id=\"fs-id1165137428180\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{tan}\\,\\theta =\\frac{-6}{6}=-1\u21d2\\theta ={\\mathrm{tan}}^{-1}\\left(-1\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-45\u00b0\\hfill \\end{array}[\/latex]<\/div>\nHowever, the angle terminates in the fourth quadrant, so we add 360\u00b0 to obtain a positive angle. Thus,[latex]\\,-45\u00b0+360\u00b0=315\u00b0.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_08_018\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152915\/CNX_Precalc_Figure_08_08_018.jpg\" alt=\"Plot of the position vector extending into Q4 from the origin with the magnitude 6rad2.\" width=\"487\" height=\"316\"> <strong>Figure 6.<\/strong>[\/caption]\n<p id=\"fs-id1165134254080\">[\/hidden-answer]<span id=\"fs-id1165134196104\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135536154\">\n<div id=\"fs-id1165135536156\">\n<h3>Showing That Two Vectors Are Equal<\/h3>\n<p id=\"fs-id1165135349355\">Show that vector <strong><em>v<\/em><\/strong> with <span class=\"no-emphasis\">initial point<\/span> at[latex]\\,\\left(5,-3\\right)\\,[\/latex]and <span class=\"no-emphasis\">terminal point<\/span> at[latex]\\,\\left(-1,2\\right)\\,[\/latex]is equal to vector <strong><em>u<\/em><\/strong> with initial point at[latex]\\,\\left(-1,-3\\right)\\,[\/latex]and terminal point at[latex]\\,\\left(-7,2\\right).\\,[\/latex]Draw the position vector on the same grid as <strong><em>v<\/em><\/strong> and <strong><em>u<\/em><\/strong>. Next, find the magnitude and direction of each vector.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137431395\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137431395\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137431395\"]\n<p id=\"fs-id1165137431397\">As shown in <a class=\"autogenerated-content\" href=\"#Figure_08_08_005\">(Figure)<\/a>, draw the vector[latex]\\,v\\,[\/latex]starting at initial[latex]\\,\\left(5,-3\\right)\\,[\/latex]and terminal point[latex]\\,\\left(-1,2\\right).\\,[\/latex]Draw the vector[latex]\\,u\\,[\/latex]with initial point[latex]\\,\\left(-1,-3\\right)\\,[\/latex]and terminal point[latex]\\,\\left(-7,2\\right).\\,[\/latex]Find the standard position for each.<\/p>\n<p id=\"fs-id1165134314743\">Next, find and sketch the position vector for <strong><em>v<\/em><\/strong> and <strong><em>u<\/em><\/strong>. We have<\/p>\n\n<div id=\"fs-id1165137557695\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u2329-1-5,2-\\left(-3\\right)\u232a\\hfill \\\\ \\text{ }=\u2329-6,5\u232a\\hfill \\\\ \\hfill \\\\ u=\u2329-7-\\left(-1\\right),2-\\left(-3\\right)\u232a\\hfill \\\\ \\text{ }=\u2329-6,5\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135638538\">Since the position vectors are the same, <strong><em>v<\/em><\/strong> and <strong><em>u<\/em><\/strong> are the same.<\/p>\n<p id=\"fs-id1165135353002\">An alternative way to check for vector equality is to show that the magnitude and direction are the same for both vectors. To show that the magnitudes are equal, use the Pythagorean Theorem.<\/p>\n\n<div id=\"fs-id1165134224057\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|v|=\\sqrt{{\\left(-1-5\\right)}^{2}+{\\left(2-\\left(-3\\right)\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{{\\left(-6\\right)}^{2}+{\\left(5\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{36+25}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{61}\\hfill \\\\ |u|=\\sqrt{{\\left(-7-\\left(-1\\right)\\right)}^{2}+{\\left(2-\\left(-3\\right)\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{{\\left(-6\\right)}^{2}+{\\left(5\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{36+25}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{61}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135194193\">As the magnitudes are equal, we now need to verify the direction. Using the tangent function with the position vector gives<\/p>\n\n<div id=\"fs-id1165134339912\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{tan}\\,\\theta =-\\frac{5}{6}\u21d2\\theta ={\\mathrm{tan}}^{-1}\\left(-\\frac{5}{6}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-39.8\u00b0\\hfill \\end{array}[\/latex]<\/div>\nHowever, we can see that the position vector terminates in the second quadrant, so we add[latex]\\,180\u00b0.\\,[\/latex]Thus, the direction is[latex]\\,-39.8\u00b0+180\u00b0=140.2\u00b0.[\/latex]\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152927\/CNX_Precalc_Figure_08_08_005n.jpg\" alt=\"Plot of the two given vectors their same position vector.\" width=\"487\" height=\"440\"> <strong>Figure 7.<\/strong>[\/caption]\n<p id=\"fs-id1165134569140\">[\/hidden-answer]<span id=\"fs-id1165134108430\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137806748\" class=\"bc-section section\">\n<h3>Performing Vector Addition and Scalar Multiplication<\/h3>\nNow that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector <strong>[latex]u[\/latex]<\/strong>[latex]=\u2329x,y\u232a[\/latex]as an arrow or directed line segment from the origin to the point[latex]\\,\\left(x,y\\right),\\,[\/latex]vectors can be situated anywhere in the plane. The sum of two vectors <strong><em>u<\/em><\/strong> and <strong><em>v<\/em><\/strong>, or vector addition, produces a third vector <strong><em>u<\/em><\/strong>+ <strong><em>v<\/em><\/strong>, the resultant vector.\n<p id=\"fs-id1165135609392\">To find <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, we first draw the vector <strong><em>u<\/em><\/strong>, and from the terminal end of <strong><em>u<\/em><\/strong>, we drawn the vector <strong><em>v<\/em><\/strong>. In other words, we have the initial point of <strong><em>v<\/em><\/strong> meet the terminal end of <strong><em>u<\/em><\/strong>. This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong> is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of <strong><em>u<\/em><\/strong> to the end of <strong><em>v<\/em><\/strong> in a straight path, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_08_008\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_08_008\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152941\/CNX_Precalc_Figure_08_08_008.jpg\" alt=\"Diagrams of vector addition and subtraction.\" width=\"487\" height=\"149\"> <strong>Figure 8.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135415868\">Vector subtraction is similar to vector addition. To find <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>, view it as <strong><em>u<\/em><\/strong> + (\u2212<strong><em>v<\/em><\/strong>). Adding \u2212<strong><em>v<\/em><\/strong> is reversing direction of <strong><em>v<\/em><\/strong> and adding it to the end of <strong><em>u<\/em><\/strong>. The new vector begins at the start of <strong><em>u<\/em><\/strong> and stops at the end point of \u2212<strong><em>v<\/em><\/strong>. See <a class=\"autogenerated-content\" href=\"#Figure_08_08_009\">(Figure)<\/a> for a visual that compares vector addition and vector subtraction using <span class=\"no-emphasis\">parallelograms<\/span>.<\/p>\n\n<div id=\"Figure_08_08_009\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152958\/CNX_Precalc_Figure_08_08_009.jpg\" alt=\"Showing vector addition and subtraction with parallelograms. For addition, the base is u, the side is v, the diagonal connecting the start of the base to the end of the side is u+v. For subtraction, thetop is u, the side is -v, and the diagonal connecting the start of the top to the end of the side is u-v.\" width=\"487\" height=\"128\"> <strong>Figure 9.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"Example_08_08_05\" class=\"textbox examples\">\n<div id=\"fs-id1165135410202\">\n<div id=\"fs-id1165135410204\">\n<h3>Adding and Subtracting Vectors<\/h3>\n<p id=\"fs-id1165135410210\">Given <strong>[latex]u[\/latex]<\/strong>[latex]=\u23293,-2\u232a[\/latex]and<strong>[latex]v[\/latex]<\/strong>[latex]=\u2329-1,4\u232a,[\/latex]find two new vectors <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, and <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135407415\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135407415\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135407415\"]\n<p id=\"fs-id1165135407417\">To find the sum of two vectors, we add the components. Thus,<\/p>\n\n<div id=\"fs-id1165135407421\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}u+v=\u23293,-2\u232a+\u2329-1,4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\u23293+\\left(-1\\right),-2+4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\u23292,2\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134081345\">See <a class=\"autogenerated-content\" href=\"#Figure_08_08_019\">(Figure)<\/a><strong>(a)<\/strong>.<\/p>\n<p id=\"fs-id1165134081354\">To find the difference of two vectors, add the negative components of<strong>[latex]\\,v\\,[\/latex]<\/strong>to<strong>[latex]\\,u.\\,[\/latex]<\/strong>Thus,<\/p>\n\n<div id=\"fs-id1165137832939\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}u+\\left(-v\\right)=\u23293,-2\u232a+\u23291,-4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\u23293+1,-2+\\left(-4\\right)\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\u23294,-6\u232a\\hfill \\end{array}[\/latex]<\/div>\nSee <a class=\"autogenerated-content\" href=\"#Figure_08_08_019\">(Figure)<\/a><strong>(b).<\/strong>\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153012\/CNX_Precalc_Figure_08_08_019.jpg\" alt=\"Further diagrams of vector addition and subtraction.\" width=\"731\" height=\"292\"> <strong>Figure 10. <\/strong>(a) Sum of two vectors (b) Difference of two vectors[\/caption]\n<p id=\"fs-id1165134179704\">[\/hidden-answer]<\/p>\n\n<div id=\"Figure_08_08_019\" class=\"medium\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137589359\" class=\"bc-section section\">\n<h3>Multiplying By a Scalar<\/h3>\n<p id=\"fs-id1165135472137\">While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.<\/p>\n\n<div id=\"fs-id1165134467706\">\n<h3>Scalar Multiplication<\/h3>\n<p id=\"fs-id1165135316119\">Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply <strong>[latex]v[\/latex]<\/strong>[latex]=\u2329a,b\u232a[\/latex] by [latex]k[\/latex], we have<\/p>\n\n<div id=\"fs-id1165133278770\" class=\"unnumbered aligncenter\">[latex]kv=\u2329ka,kb\u232a[\/latex]<\/div>\n<p id=\"fs-id1165137681116\">Only the magnitude changes, unless[latex]\\,k\\,[\/latex]is negative, and then the vector reverses direction.<\/p>\n\n<\/div>\n<div id=\"Example_08_08_06\" class=\"textbox examples\">\n<div id=\"fs-id1165134294910\">\n<div id=\"fs-id1165134294912\">\n<h3>Performing Scalar Multiplication<\/h3>\n<p id=\"fs-id1165135315522\">Given vector<strong>[latex]\\,v[\/latex]<\/strong>[latex]=\u23293,1\u232a,\\,[\/latex]find 3<strong><em>v<\/em><\/strong>, [latex]\\frac{1}{2}[\/latex]<strong>[latex]v,\\,[\/latex]<\/strong>and \u2212<strong><em>v<\/em><\/strong>.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135580319\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135580319\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135580319\"]\n<p id=\"fs-id1165135580321\">See <a class=\"autogenerated-content\" href=\"#Figure_08_08_007\">(Figure)<\/a> for a geometric interpretation. If<strong>[latex]\\,v[\/latex]<\/strong>[latex]=\u23293,1\u232a,[\/latex]then<\/p>\n\n<div id=\"eip-id3684825\" class=\"unnumbered\">[latex]\\begin{array}{l}\\,\\,3v=\u23293\\cdot 3,3\\cdot 1\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,=\u23299,3\u232a\\hfill \\\\ \\,\\frac{1}{2}v=\u2329\\frac{1}{2}\\cdot 3,\\frac{1}{2}\\cdot 1\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,=\u2329\\frac{3}{2},\\frac{1}{2}\u232a\\hfill \\\\ -v=\u2329-3,-1\u232a\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_08_08_007\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153015\/CNX_Precalc_Figure_08_08_007.jpg\" alt=\"Showing the effect of scaling a vector: 3x, 1x, .5x, and -1x. The 3x is three times as long, the 1x stays the same, the .5x halves the length, and the -1x reverses the direction of the vector but keeps the length the same. The rest keep the same direction; only the magnitude changes.\" width=\"487\" height=\"367\"> <strong>Figure 11.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134361434\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134153304\">Notice that the vector 3<strong><em>v<\/em><\/strong> is three times the length of <strong><em>v<\/em><\/strong>, [latex]\\frac{1}{2}[\/latex]<strong>[latex]v\\,[\/latex]<\/strong>is half the length of <strong><em>v<\/em><\/strong>, and \u2013<strong><em>v<\/em><\/strong> is the same length of <strong><em>v<\/em><\/strong>, but in the opposite direction.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165134159656\">\n<div id=\"fs-id1165134159657\">\n<p id=\"fs-id1165134159658\">Find the <span class=\"no-emphasis\">scalar multiple<\/span> 3<strong>[latex]u[\/latex]<\/strong> given <strong>[latex]u[\/latex]<\/strong>[latex]=\u23295,4\u232a.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135367782\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135367782\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135367782\"]\n<p id=\"fs-id1165135367783\">[latex]3u=\u232915,12\u232a[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_07\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165135650641\">\n<h3>Using Vector Addition and Scalar Multiplication to Find a New Vector<\/h3>\n<p id=\"fs-id1165135434035\">Given <strong>[latex]u[\/latex]<\/strong>[latex]=\u23293,-2\u232a[\/latex]and<strong>[latex]v[\/latex]<\/strong>[latex]=\u2329-1,4\u232a,[\/latex]find a new vector <strong><em>w<\/em><\/strong> = 3<strong><em>u<\/em><\/strong> + 2<strong><em>v<\/em><\/strong>.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137451791\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137451791\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137451791\"]First, we must multiply each vector by the scalar.\n<div id=\"fs-id1165137451796\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}3u=3\u23293,-2\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\u23299,-6\u232a\\hfill \\\\ 2v=2\u2329-1,4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\u2329-2,8\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135208894\">Then, add the two together.<\/p>\n\n<div id=\"fs-id1165135208898\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}w=3u+2v\\hfill \\\\ \\,\\,\\,\\,\\,=\u23299,-6\u232a+\u2329-2,8\u232a\\hfill \\\\ \\,\\,\\,\\,\\,=\u23299-2,-6+8\u232a\\hfill \\\\ \\,\\,\\,\\,\\,=\u23297,2\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134066537\">So, <strong>[latex]w[\/latex]<\/strong>[latex]=\u23297,2\u232a.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137922516\" class=\"bc-section section\">\n<h3>Finding Component Form<\/h3>\n<p id=\"fs-id1165137401307\">In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the[latex]\\,x\\,[\/latex]direction, and the vertical component is the[latex]\\,y\\,[\/latex]direction. For example, we can see in the graph in <a class=\"autogenerated-content\" href=\"#Figure_08_08_020\">(Figure)<\/a> that the position vector[latex]\u23292,3\u232a[\/latex]comes from adding the vectors <strong><em>v<\/em><\/strong><sub>1<\/sub> and <strong><em>v<\/em><\/strong><sub>2<\/sub>. We have <strong><em>v<\/em><\/strong><sub>1<\/sub> with initial point[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminal point[latex]\\,\\left(2,0\\right).\\,[\/latex]<\/p>\n\n<div id=\"fs-id1165133045345\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{v}_{1}=\u23292-0,0-0\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\u23292,0\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135440228\">We also have <strong><em>v<\/em><\/strong><sub>2<\/sub> with initial point[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminal point[latex]\\,\\left(0,\\,3\\right).\\,[\/latex]<\/p>\n\n<div id=\"eip-id4003269\" class=\"unnumbered\">[latex]\\begin{array}{l}{v}_{2}=\u23290-0,3-0\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,=\u23290,3\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135457923\">Therefore, the position vector is<\/p>\n\n<div id=\"fs-id1165135457926\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u23292+0,3+0\u232a\\hfill \\\\ \\,\\,\\,=\u23292,3\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137400265\">Using the Pythagorean Theorem, the magnitude of <strong><em>v<\/em><\/strong><sub>1<\/sub> is 2, and the magnitude of <strong><em>v<\/em><\/strong><sub>2<\/sub> is 3. To find the magnitude of <strong><em>v<\/em><\/strong>, use the formula with the position vector.<\/p>\n\n<div id=\"fs-id1165137849225\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}|v|=\\sqrt{|{v}_{1}{|}^{2}+|{v}_{2}{|}^{2}}\\hfill \\\\ \\begin{array}{l}\\,\\,\\,\\,\\,=\\sqrt{{2}^{2}+{3}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,=\\sqrt{13}\\hfill \\end{array}\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137387521\">The magnitude of <strong><em>v<\/em><\/strong> is[latex]\\,\\sqrt{13}.\\,[\/latex]To find the direction, we use the tangent function[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{y}{x}.[\/latex]<\/p>\n\n<div id=\"fs-id1165134224497\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{tan}\\,\\theta =\\frac{{v}_{2}}{{v}_{1}}\\hfill \\\\ \\mathrm{tan}\\,\\theta =\\frac{3}{2}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\theta ={\\mathrm{tan}}^{-1}\\left(\\frac{3}{2}\\right)=56.3\u00b0\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_08_08_020\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153029\/CNX_Precalc_Figure_08_08_020.jpg\" alt=\"Diagram of a vector in root position with its horizontal and vertical components.\" width=\"487\" height=\"289\"> <strong>Figure 12.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165134154507\">Thus, the magnitude of<strong>[latex]\\,v\\,[\/latex]<\/strong>is[latex]\\,\\sqrt{13}\\,[\/latex]and the direction is[latex]\\,{56.3}^{\\circ }[\/latex]off the horizontal.<\/p>\n\n<div id=\"Example_08_08_08\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137761654\">\n<h3>Finding the Components of the Vector<\/h3>\n<p id=\"fs-id1165134295643\">Find the components of the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>with initial point[latex]\\,\\left(3,2\\right)\\,[\/latex]and terminal point[latex]\\,\\left(7,4\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135253181\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135253181\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135253181\"]\n<p id=\"fs-id1165135253183\">First find the standard position.<\/p>\n\n<div id=\"fs-id1165135253186\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u23297-3,4-2\u232a\\hfill \\\\ \\,\\,\\,=\u23294,2\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137827401\">See the illustration in <a class=\"autogenerated-content\" href=\"#Figure_08_08_021\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_08_021\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153036\/CNX_Precalc_Figure_08_08_021.jpg\" alt=\"Diagram of a vector in root position with its horizontal (4,0) and vertical (0,2) components.\" width=\"487\" height=\"254\"> <strong>Figure 13.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137437225\">The horizontal component is <strong>[latex]{v}_{1}[\/latex]<\/strong>[latex]=\u23294,0\u232a\\,[\/latex]and the vertical component is<strong>[latex]\\,{v}_{2}[\/latex]<\/strong>[latex]=\u23290,2\u232a.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133299033\" class=\"bc-section section\">\n<h3>Finding the Unit Vector in the Direction of <em>v<\/em><\/h3>\n<p id=\"fs-id1165135634115\">In addition to finding a vector\u2019s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then preserve the direction of the original vector while simplifying calculations.<\/p>\n<p id=\"fs-id1165135190463\">Unit vectors are defined in terms of components. The horizontal unit vector is written as <strong>[latex]i[\/latex]<\/strong>[latex]=\u23291,0\u232a[\/latex]and is directed along the positive horizontal axis. The vertical unit vector is written as<strong>[latex]j[\/latex]<\/strong>[latex]=\u23290,1\u232a[\/latex]and is directed along the positive vertical axis. See <a class=\"autogenerated-content\" href=\"#Figure_08_08_011\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_08_011\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153043\/CNX_Precalc_Figure_08_08_011n.jpg\" alt=\"Plot showing the unit vectors i=91,0) and j=(0,1)\" width=\"487\" height=\"253\"> <strong>Figure 14.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165137838847\">\n<h3>The Unit Vectors<\/h3>\n<p id=\"fs-id1165134393822\">If<strong>[latex]\\,v\\,[\/latex]<\/strong>is a nonzero vector, then<strong>[latex]\\,\\frac{v}{|v|}\\,[\/latex]<\/strong>is a unit vector in the direction of<strong>[latex]\\,v.\\,[\/latex]<\/strong>Any vector divided by its magnitude is a unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.<\/p>\n\n<\/div>\n<div id=\"Example_08_08_09\" class=\"textbox examples\">\n<div id=\"fs-id1165137423314\">\n<div id=\"fs-id1165137423316\">\n<h3>Finding the Unit Vector in the Direction of <em>v<\/em><\/h3>\n<p id=\"fs-id1165134058261\">Find a unit vector in the same direction as <strong>[latex]v[\/latex]<\/strong>[latex]=\u2329-5,12\u232a.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137900009\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137900009\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137900009\"]\n<p id=\"fs-id1165137900011\">First, we will find the magnitude.<\/p>\n\n<div id=\"fs-id1165137900014\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|v|=\\sqrt{{\\left(-5\\right)}^{2}+{\\left(12\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{25+144}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{169}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=13\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135570479\">Then we divide each component by[latex]\\,|v|,\\,[\/latex]which gives a unit vector in the same direction as <strong><em>v<\/em><\/strong>:<\/p>\n\n<div id=\"fs-id1165135193310\" class=\"unnumbered aligncenter\">[latex]\\frac{v}{|v|}=-\\frac{5}{13}i+\\frac{12}{13}j[\/latex]<\/div>\nor, in component form\n<div id=\"fs-id1165137653189\" class=\"unnumbered aligncenter\">[latex]\\frac{v}{|v|}=\u2329-\\frac{5}{13},\\frac{12}{13}\u232a[\/latex]<\/div>\n<p id=\"fs-id1165135154225\">See <a class=\"autogenerated-content\" href=\"#Figure_08_08_012\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_08_012\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153046\/CNX_Precalc_Figure_08_08_012.jpg\" alt=\"Plot showing the unit vector (-5\/13, 12\/13) in the direction of (-5, 12)\" width=\"487\" height=\"628\"> <strong>Figure 15.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137921617\">Verify that the magnitude of the unit vector equals 1. The magnitude of[latex]\\,-\\frac{5}{13}i+\\frac{12}{13}j\\,[\/latex]is given as<\/p>\n\n<div id=\"fs-id1165132960731\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\sqrt{{\\left(-\\frac{5}{13}\\right)}^{2}+{\\left(\\frac{12}{13}\\right)}^{2}}=\\sqrt{\\frac{25}{169}+\\frac{144}{169}}\\hfill \\\\ \\text{ }=\\sqrt{\\frac{169}{169}}=1\\hfill \\end{array}[\/latex]<\/div>\nThe vector <strong><em>u<\/em><\/strong>[latex]=\\frac{5}{13}[\/latex]<strong><em>i<\/em><\/strong>[latex]+\\frac{12}{13}[\/latex]<strong><em>j<\/em><\/strong> is the unit vector in the same direction as <strong><em>v<\/em><\/strong>[latex]=\u2329-5,12\u232a.[\/latex][\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137692582\" class=\"bc-section section\">\n<h3>Performing Operations with Vectors in Terms of <em>i<\/em> and <em>j <\/em><\/h3>\n<p id=\"fs-id1165135333199\">So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of <strong><em>i<\/em><\/strong> and <strong><em>j<\/em><\/strong>.<\/p>\n\n<div id=\"fs-id1165133075623\">\n<h3>Vectors in the Rectangular Plane<\/h3>\n<p id=\"fs-id1165133075632\">Given a vector<strong>[latex]\\,v\\,[\/latex]<\/strong>with initial point[latex]\\,P=\\left({x}_{1},{y}_{1}\\right)\\,[\/latex]and terminal point [latex]Q=\\left({x}_{2},{y}_{2}\\right),[\/latex] <strong><em>v<\/em><\/strong> is written as<\/p>\n\n<div id=\"fs-id1165135434061\" class=\"unnumbered aligncenter\">[latex]v=\\left({x}_{2}-{x}_{1}\\right)i+\\left({y}_{2}-{y}_{1}\\right)j[\/latex]<\/div>\n<p id=\"fs-id1165135180021\">The position vector from[latex]\\,\\left(0,0\\right)\\,[\/latex]to[latex]\\,\\left(a,b\\right),\\,[\/latex]where[latex]\\,\\left({x}_{2}-{x}_{1}\\right)=a\\,[\/latex]and[latex]\\,\\left({y}_{2}-{y}_{1}\\right)=b,\\,[\/latex]is written as <strong><em>v<\/em><\/strong> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em>. This vector sum is called a linear combination of the vectors <strong><em>i<\/em><\/strong> and <strong><em>j<\/em><\/strong>.<\/p>\n<p id=\"fs-id1165134151128\">The magnitude of <strong><em>v<\/em><\/strong> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> is given as[latex]\\,|v|=\\sqrt{{a}^{2}+{b}^{2}}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_08_010\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_08_010\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153051\/CNX_Precalc_Figure_08_08_010new.jpg\" alt=\"Plot showing vectors in rectangular coordinates in terms of i and j. The position vector v (in orange) extends from the origin to some point (a,b) in Q1. The horizontal (ai) and vertical (bj) components are shown.\" width=\"487\" height=\"237\"> <strong>Figure 16.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"Example_08_08_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137640998\">\n<div id=\"fs-id1165137398636\">\n<h3>Writing a Vector in Terms of <em>i<\/em> and <em>j<\/em><\/h3>\n<p id=\"fs-id1165135195612\">Given a vector<strong>[latex]\\,v\\,[\/latex]<\/strong>with initial point[latex]\\,P=\\left(2,-6\\right)\\,[\/latex]and terminal point[latex]\\,Q=\\left(-6,6\\right),\\,[\/latex]write the vector in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.[\/latex]<\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165137415659\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137415659\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137415659\"]\n<p id=\"fs-id1165137415662\">Begin by writing the general form of the vector. Then replace the coordinates with the given values.<\/p>\n\n<div id=\"fs-id1165137415666\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\\left({x}_{2}-{x}_{1}\\right)i+\\left({y}_{2}-{y}_{1}\\right)j\\hfill \\\\ \\,\\,\\,=\\left(-6-2\\right)i+\\left(6-\\left(-6\\right)\\right)j\\hfill \\\\ \\,\\,\\,=-8i+12j\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_11\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165134130103\">\n<h3>Writing a Vector in Terms of <em>i<\/em> and <em>j<\/em> Using Initial and Terminal Points<\/h3>\n<p id=\"fs-id1165137653576\">Given initial point[latex]\\,{P}_{1}=\\left(-1,3\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(2,7\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165132036992\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132036992\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165132036992\"]\n<p id=\"fs-id1165132036994\">Begin by writing the general form of the vector. Then replace the coordinates with the given values.<\/p>\n\n<div id=\"fs-id1165132036998\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\\left({x}_{2}-{x}_{1}\\right)i+\\left({y}_{2}-{y}_{1}\\right)j\\hfill \\\\ v=\\left(2-\\left(-1\\right)\\right)i+\\left(7-3\\right)j\\hfill \\\\ \\,\\,=3i+4j\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134309524\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_08_02\">\n<div id=\"fs-id1165133015819\">\n<p id=\"fs-id1165133015820\">Write the vector<strong>[latex]\\,u\\,[\/latex]<\/strong>with initial point[latex]\\,P=\\left(-1,6\\right)\\,[\/latex]and terminal point[latex]\\,Q=\\left(7,-5\\right)\\,[\/latex]in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.[\/latex]<\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165137726280\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137726280\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137726280\"]\n<p id=\"fs-id1165137726281\">[latex]u=8i-11j[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134148227\" class=\"bc-section section\">\n<h3>Performing Operations on Vectors in Terms of <em>i<\/em> and <em>j<\/em><\/h3>\n<p id=\"fs-id1165135610243\">When vectors are written in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j,\\,[\/latex]<\/strong>we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.<\/p>\n\n<div id=\"fs-id1165135452303\">\n<h3>Adding and Subtracting Vectors in Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165137725223\">Given <strong><em>v<\/em><\/strong> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> and <strong><em>u<\/em><\/strong> = <em>c<strong>i<\/strong><\/em> + <em>d<strong>j<\/strong><\/em>, then<\/p>\n\n<div id=\"fs-id1165135332191\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}v+u=\\left(a+c\\right)i+\\left(b+d\\right)j\\\\ v-u=\\left(a-c\\right)i+\\left(b-d\\right)j\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_08_08_12\" class=\"textbox examples\">\n<div id=\"fs-id1165135481247\">\n<div id=\"fs-id1165135481249\">\n<h3>Finding the Sum of the Vectors<\/h3>\n<p id=\"fs-id1165135204368\">Find the sum of[latex]\\,{v}_{1}=2i-3j\\,[\/latex]and[latex]\\,{v}_{2}=4i+5j.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134547417\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134547417\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134547417\"]\n<p id=\"fs-id1165134547419\">According to the formula, we have<\/p>\n\n<div id=\"fs-id1165135516771\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{v}_{1}+{v}_{2}=\\left(2+4\\right)i+\\left(-3+5\\right)j\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=6i+2j\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h3>Calculating the Component Form of a Vector: Direction<\/h3>\n<p id=\"fs-id1165135616272\">We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using[latex]\\,i\\,\\,\\text{and}\\,\\,j.\\,[\/latex]For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.<\/p>\n<p id=\"fs-id1165135515850\">Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with<strong>[latex]\\,|v|\\,[\/latex]<\/strong>replacing<strong>[latex]\\,r.[\/latex]<\/strong><\/p>\n\n<div id=\"fs-id1165134486704\">\n<h3>Vector Components in Terms of Magnitude and Direction<\/h3>\n<p id=\"fs-id1165137848784\">Given a position vector[latex]\\,v=\u2329x,y\u232a\\,[\/latex]and a direction angle[latex]\\,\\theta ,[\/latex]<\/p>\n\n<div id=\"fs-id1165133354263\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{lll}\\mathrm{cos}\\,\\theta =\\frac{x}{|v|}\\hfill &amp; \\text{and}\\begin{array}{cc}&amp; \\end{array}\\hfill &amp; \\mathrm{sin}\\,\\theta =\\frac{y}{|v|}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,x=|v|\\mathrm{cos}\\,\\theta \\begin{array}{cc}&amp; \\end{array}\\hfill &amp; \\hfill &amp; \\,\\,\\,\\,\\,\\,\\,y=|v|\\mathrm{sin}\\,\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135554288\">Thus,[latex]\\,v=xi+yj=|v|\\mathrm{cos}\\,\\theta i+|v|\\mathrm{sin}\\,\\theta j,\\,[\/latex]and magnitude is expressed as[latex]\\,|v|=\\sqrt{{x}^{2}+{y}^{2}}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"Example_08_08_13\" class=\"textbox examples\">\n<div id=\"fs-id1165134280403\">\n<div id=\"fs-id1165134280405\">\n<h3>Writing a Vector in Terms of Magnitude and Direction<\/h3>\n<p id=\"fs-id1165134280410\">Write a vector with length 7 at an angle of 135\u00b0 to the positive\n<em>x<\/em>-axis in terms of magnitude and direction.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135421536\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135421536\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135421536\"]\n<p id=\"fs-id1165135421538\">Using the conversion formulas[latex]\\,x=|v|\\mathrm{cos}\\,\\theta i\\,[\/latex]and[latex]\\,y=|v|\\mathrm{sin}\\,\\theta j,\\,[\/latex]we find that<\/p>\n\n<div id=\"fs-id1165135512708\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x=7\\mathrm{cos}\\left(135\u00b0\\right)i\\hfill \\\\ \\,\\,\\,=-\\frac{7\\sqrt{2}}{2}\\hfill \\\\ y=7\\mathrm{sin}\\left(135\u00b0\\right)j\\hfill \\\\ \\,\\,\\,=\\frac{7\\sqrt{2}}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165133389128\">This vector can be written as[latex]\\,v=7\\mathrm{cos}\\left(135\u00b0\\right)i+7\\mathrm{sin}\\left(135\u00b0\\right)j\\,[\/latex]or simplified as<\/p>\n\n<div id=\"fs-id1165135515872\" class=\"unnumbered aligncenter\">[latex]v=-\\frac{7\\sqrt{2}}{2}i+\\frac{7\\sqrt{2}}{2}j[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133095102\" class=\"textbox tryit\">\n<div id=\"ti_08_08_03\">\n<div id=\"fs-id1165133306703\">\n<p id=\"fs-id1165133306704\">A vector travels from the origin to the point[latex]\\,\\left(3,5\\right).\\,[\/latex]Write the vector in terms of magnitude and direction.<\/p>\n\n<\/div>\n<div id=\"fs-id1165131968594\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131968594\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165131968594\"]\n<p id=\"fs-id1165134391587\">[latex]v=\\sqrt{34}\\mathrm{cos}\\left(59\u00b0\\right)i+\\sqrt{34}\\mathrm{sin}\\left(59\u00b0\\right)j[\/latex]<\/p>\nMagnitude =[latex]\\,\\sqrt{34}[\/latex]\n<p id=\"fs-id1165135436242\">[latex]\\theta ={\\mathrm{tan}}^{-1}\\left(\\frac{5}{3}\\right)=59.04\u00b0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135381330\" class=\"bc-section section\">\n<h3>Finding the Dot Product of Two Vectors<\/h3>\n<p id=\"fs-id1165135381335\">As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the <em>dot product<\/em> and the <em>cross product<\/em>. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.<\/p>\n<p id=\"fs-id1165134085621\">The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.<\/p>\n\n<div id=\"fs-id1165134085624\">\n<h3>Dot Product<\/h3>\n<p id=\"fs-id1165134129884\">The dot product of two vectors[latex]\\,v=\u2329a,b\u232a\\,[\/latex]and[latex]\\,u=\u2329c,d\u232a\\,[\/latex]is the sum of the product of the horizontal components and the product of the vertical components.<\/p>\n\n<div id=\"fs-id1165137656787\" class=\"unnumbered aligncenter\">[latex]v\\cdot u=ac+bd[\/latex]<\/div>\nTo find the angle between the two vectors, use the formula below.\n<div id=\"fs-id1165134156064\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cos}\\,\\theta =\\frac{v}{|v|}\\cdot \\frac{u}{|u|}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_08_08_14\" class=\"textbox examples\">\n<div id=\"fs-id1165135440032\">\n<div id=\"fs-id1165134149775\">\n<h3>Finding the Dot Product of Two Vectors<\/h3>\nFind the dot product of<strong>[latex]\\,v=\u23295,12\u232a\\,[\/latex]<\/strong>and<strong>[latex]\\,u=\u2329-3,4\u232a.[\/latex]<\/strong>\n\n<\/div>\n<div id=\"fs-id1165137863042\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137863042\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137863042\"]\n<p id=\"fs-id1165137863044\">Using the formula, we have<\/p>\n\n<div id=\"fs-id1165137863047\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v\\cdot u=\u23295,12\u232a\\cdot \u2329-3,4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5\\cdot \\left(-3\\right)+12\\cdot 4\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-15+48\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=33\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_15\" class=\"textbox examples\">\n<div id=\"fs-id1165134192846\">\n<div id=\"fs-id1165134192848\">\n<h3>Finding the Dot Product of Two Vectors and the Angle between Them<\/h3>\n<p id=\"fs-id1165134192853\">Find the dot product of <strong><em>v<\/em><\/strong><sub>1<\/sub> = 5<strong><em>i<\/em><\/strong> + 2<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong><sub>2<\/sub> = 3<strong><em>i<\/em><\/strong> + 7<strong><em>j<\/em><\/strong>. Then, find the angle between the two vectors.<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165134330343\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134330343\"]\n<p id=\"fs-id1165134330343\">Finding the dot product, we multiply corresponding components.<\/p>\n\n<div id=\"fs-id1165134330347\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{v}_{1}\\cdot {v}_{2}=\u23295,2\u232a\\cdot \u23293,7\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5\\cdot 3+2\\cdot 7\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=15+14\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=29\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135417616\">To find the angle between them, we use the formula[latex]\\,\\mathrm{cos}\\,\\theta =\\frac{v}{|v|}\\cdot \\frac{u}{|u|}.[\/latex]<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{v}{|v|}\\cdot \\frac{u}{|u|}=\u2329\\frac{5}{\\sqrt{29}}+\\frac{2}{\\sqrt{29}}\u232a\\cdot \u2329\\frac{3}{\\sqrt{58}}+\\frac{7}{\\sqrt{58}}\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{5}{\\sqrt{29}}\\cdot \\frac{3}{\\sqrt{58}}+\\frac{2}{\\sqrt{29}}\\cdot \\frac{7}{\\sqrt{58}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{15}{\\sqrt{1682}}+\\frac{14}{\\sqrt{1682}}=\\frac{29}{\\sqrt{1682}}\\hfill \\\\ \\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0.707107\\hfill \\\\ {\\mathrm{cos}}^{-1}\\left(0.707107\\right)=45\u00b0\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\nSee <a class=\"autogenerated-content\" href=\"#Figure_08_08_014\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153113\/CNX_Precalc_Figure_08_08_014_Errata.jpg\" alt=\"Plot showing the two position vectors (3,7) and (5,2) and the 45 degree angle between them.\" width=\"488\" height=\"403\"> <strong>Figure 17.<\/strong>[\/caption]\n<p id=\"fs-id1165135181801\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_16\" class=\"textbox examples\">\n<div id=\"fs-id1165134041276\">\n<div id=\"fs-id1165134041278\">\n<h3>Finding the Angle between Two Vectors<\/h3>\n<p id=\"fs-id1165134041283\">Find the angle between[latex]\\,u=\u2329-3,4\u232a\\,[\/latex]and[latex]\\,v=\u23295,12\u232a.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135502770\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135502770\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135502770\"]\n<p id=\"fs-id1165135502772\">Using the formula, we have<\/p>\n\n<div id=\"fs-id1165135502775\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\theta ={\\mathrm{cos}}^{-1}\\left(\\frac{u}{|u|}\\cdot \\frac{v}{|v|}\\right)\\hfill \\\\ \\left(\\frac{u}{|u|}\\cdot \\frac{v}{|v|}\\right)=\\frac{-3i+4j}{5}\\cdot \\frac{5i+12j}{13}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\left(-\\frac{3}{5}\\cdot \\frac{5}{13}\\right)+\\left(\\frac{4}{5}\\cdot \\frac{12}{13}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-\\frac{15}{65}+\\frac{48}{65}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{33}{65}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\theta ={\\mathrm{cos}}^{-1}\\left(\\frac{33}{65}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,={59.5}^{\\circ }\\hfill \\end{array}[\/latex]<\/div>\nSee <a class=\"autogenerated-content\" href=\"#Figure_08_08_013\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153121\/CNX_Precalc_Figure_08_08_013_Errata.jpg\" alt=\"Plot showing the two position vectors (-3,4) and (5,12) and the 59.5 degree angle between them.\" width=\"488\" height=\"628\"> <strong>Figure 18.<\/strong>[\/caption]\n<p id=\"fs-id1165135442401\">[\/hidden-answer]<span id=\"fs-id1165135182874\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_17\" class=\"textbox examples\">\n<div id=\"fs-id1165137399741\">\n<div id=\"fs-id1165135503647\">\n<h3>Finding Ground Speed and Bearing Using Vectors<\/h3>\n<p id=\"fs-id1165135503653\">We now have the tools to solve the problem we introduced in the opening of the section.<\/p>\n<p id=\"fs-id1165135503656\">An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140\u00b0. A north wind (from north to south) is blowing at 16.2 miles per hour. What are the ground speed and actual bearing of the plane? See <a class=\"autogenerated-content\" href=\"#Figure_08_08_015\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_08_015\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153130\/CNX_Precalc_Figure_08_08_015.jpg\" alt=\"Image of a plan flying SE at 140 degrees and the north wind blowing.\" width=\"487\" height=\"462\"> <strong>Figure 19.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134043866\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134043866\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134043866\"]\n<p id=\"fs-id1165134043868\">The ground speed is represented by[latex]\\,x\\,[\/latex]in the diagram, and we need to find the angle[latex]\\,\\alpha \\,[\/latex]in order to calculate the adjusted bearing, which will be[latex]\\,\\,140\u00b0+\\alpha \\,.[\/latex]<\/p>\nNotice in <a class=\"autogenerated-content\" href=\"#Figure_08_08_015\">(Figure)<\/a>, that angle[latex]\\,BCO\\,[\/latex]must be equal to angle[latex]\\,AOC\\,[\/latex]by the rule of alternating interior angles, so angle[latex]\\,BCO\\,[\/latex]is 140\u00b0. We can find[latex]\\,x\\,[\/latex]by the Law of Cosines:\n<div id=\"fs-id1165135181130\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{x}^{2}={\\left(16.2\\right)}^{2}+{\\left(200\\right)}^{2}-2\\left(16.2\\right)\\left(200\\right)\\mathrm{cos}\\left(140\u00b0\\right)\\hfill \\\\ {x}^{2}=45,226.41\\hfill \\\\ \\,\\,\\,x=\\sqrt{45,226.41}\\hfill \\\\ \\,\\,\\,x=212.7\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135453031\">The ground speed is approximately 213 miles per hour. Now we can calculate the bearing using the Law of Sines.<\/p>\n\n<div id=\"fs-id1165135453035\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{\\mathrm{sin}\\,\\alpha }{16.2}=\\frac{\\mathrm{sin}\\left(140\u00b0\\right)}{212.7}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\mathrm{sin}\\,\\alpha =\\frac{16.2\\mathrm{sin}\\left(140\u00b0\\right)}{212.7}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0.04896\\hfill \\\\ {\\mathrm{sin}}^{-1}\\left(0.04896\\right)=2.8\u00b0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134158931\">Therefore, the plane has a SE bearing of 140\u00b0+2.8\u00b0=142.8\u00b0. The ground speed is 212.7 miles per hour.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135442438\" class=\"precalculus media\">\n<p id=\"fs-id1165135442444\">Access these online resources for additional instruction and practice with vectors.<\/p>\n\n<ul>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/introvectors\">Introduction to Vectors<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/vectoroperation\">Vector Operations<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/unitvector\">The Unit Vector<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137686628\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137686633\">\n \t<li>The position vector has its initial point at the origin. See <a class=\"autogenerated-content\" href=\"#Example_08_08_01\">(Figure)<\/a>.<\/li>\n \t<li>If the position vector is the same for two vectors, they are equal. See <a class=\"autogenerated-content\" href=\"#Example_08_08_02\">(Figure)<\/a>.<\/li>\n \t<li>Vectors are defined by their magnitude and direction. See <a class=\"autogenerated-content\" href=\"#Example_08_08_03\">(Figure)<\/a>.<\/li>\n \t<li>If two vectors have the same magnitude and direction, they are equal. See <a class=\"autogenerated-content\" href=\"#Example_08_08_04\">(Figure)<\/a>.<\/li>\n \t<li>Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. See <a class=\"autogenerated-content\" href=\"#Example_08_08_05\">(Figure)<\/a>.<\/li>\n \t<li>Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. See <a class=\"autogenerated-content\" href=\"#Example_08_08_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_08_07\">(Figure)<\/a>.<\/li>\n \t<li>Vectors are comprised of two components: the horizontal component along the positive <em>x<\/em>-axis, and the vertical component along the positive <em>y<\/em>-axis. See <a class=\"autogenerated-content\" href=\"#Example_08_08_08\">(Figure)<\/a>.<\/li>\n \t<li>The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.<\/li>\n \t<li>The magnitude of a vector in the rectangular coordinate system is[latex]\\,|v|=\\sqrt{{a}^{2}+{b}^{2}}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_08_09\">(Figure)<\/a><strong>.<\/strong><\/li>\n \t<li>In the rectangular coordinate system, unit vectors may be represented in terms of <strong>[latex]i[\/latex]<\/strong> and <strong>[latex]j[\/latex]<\/strong> where<strong>[latex]\\,i\\,[\/latex]<\/strong>represents the horizontal component and<strong>[latex]\\,j\\,[\/latex]<\/strong>represents the vertical component. Then, <strong><em>v<\/em><\/strong> = a<strong><em>i<\/em><\/strong> + b<strong><em>j<\/em><\/strong>\u2009 is a scalar multiple of<strong>[latex]\\,v\\,[\/latex]<\/strong>by real numbers[latex]\\,a\\,\\text{and}\\,b.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_08_10\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_08_11\">(Figure)<\/a>.<\/li>\n \t<li>Adding and subtracting vectors in terms of <em>i<\/em> and <em>j<\/em> consists of adding or subtracting corresponding coefficients of <em>i<\/em> and corresponding coefficients of <em>j<\/em>. See <a class=\"autogenerated-content\" href=\"#Example_08_08_12\">(Figure)<\/a>.<\/li>\n \t<li>A vector <em>v<\/em> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> is written in terms of magnitude and direction as[latex]\\,v=|v|\\mathrm{cos}\\,\\theta i+|v|\\mathrm{sin}\\,\\theta j.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_08_13\">(Figure)<\/a>.<\/li>\n \t<li>The dot product of two vectors is the product of the<strong>[latex]\\,i\\,[\/latex]<\/strong>terms plus the product of the<strong>[latex]\\,j\\,[\/latex]<\/strong>terms. See <a class=\"autogenerated-content\" href=\"#Example_08_08_14\">(Figure)<\/a>.<\/li>\n \t<li>We can use the dot product to find the angle between two vectors. <a class=\"autogenerated-content\" href=\"#Example_08_08_15\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_08_16\">(Figure)<\/a>.<\/li>\n \t<li>Dot products are useful for many types of physics applications. See <a class=\"autogenerated-content\" href=\"#Example_08_08_17\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135404190\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135361194\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135361200\">\n<div id=\"fs-id1165135361202\">\n<p id=\"fs-id1165135361203\">What are the characteristics of the letters that are commonly used to represent vectors?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135361206\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135361206\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135361206\"]\n<p id=\"fs-id1165135361208\">lowercase, bold letter, usually[latex]\\,u,v,w[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134328318\">\n<div>\n\nHow is a vector more specific than a line segment?\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135551174\">\n<div id=\"fs-id1165135551176\">\n<p id=\"fs-id1165135551178\">What are<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j,[\/latex]<\/strong>and what do they represent?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134279576\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134279576\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134279576\"]\n<p id=\"fs-id1165134279578\">They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279585\">\n<div id=\"fs-id1165135442412\">\n<p id=\"fs-id1165135442414\">What is component form?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135442420\">\n<div id=\"fs-id1165135442422\">\n<p id=\"fs-id1165135442424\">When a unit vector is expressed as[latex]\u2329a,b\u232a,[\/latex]which letter is the coefficient of the<strong>[latex]\\,i\\,[\/latex]<\/strong>and which the<strong>[latex]\\,j?[\/latex]<\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165135309849\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135309849\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135309849\"]\n<p id=\"fs-id1165135309851\">The first number always represents the coefficient of the[latex]\\,i,\\,[\/latex]and the second represents the[latex]\\,j.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134177056\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id1165137806553\">\n<div id=\"fs-id1165137806556\">\n<p id=\"fs-id1165137806558\">Given a vector with initial point[latex]\\,\\left(5,2\\right)\\,[\/latex]and terminal point[latex]\\,\\left(-1,-3\\right),\\,[\/latex]find an equivalent vector whose initial point is[latex]\\,\\left(0,0\\right).\\,[\/latex]Write the vector in component form[latex]\u2329a,b\u232a.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134318806\">\n<div id=\"fs-id1165134318807\">\n<p id=\"fs-id1165134318808\">Given a vector with initial point[latex]\\,\\left(-4,2\\right)\\,[\/latex]and terminal point[latex]\\,\\left(3,-3\\right),\\,[\/latex]find an equivalent vector whose initial point is[latex]\\,\\left(0,0\\right).\\,[\/latex]Write the vector in component form[latex]\u2329a,b\u232a.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133050481\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133050481\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133050481\"]\n<p id=\"fs-id1165133050483\">[latex]\u30087,-5\u3009[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134475649\">\n<div id=\"fs-id1165134475650\">\n<p id=\"fs-id1165134475651\">Given a vector with initial point[latex]\\,\\left(7,-1\\right)\\,[\/latex]and terminal point[latex]\\,\\left(-1,-7\\right),\\,[\/latex]find an equivalent vector whose initial point is[latex]\\,\\left(0,0\\right).\\,[\/latex]Write the vector in component form[latex]\u2329a,b\u232a.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134149962\">For the following exercises, determine whether the two vectors<strong>[latex]\\,u\\,[\/latex]<\/strong>and<strong>[latex]\\,v\\,[\/latex]<\/strong>are equal, where<strong>[latex]\\,u\\,[\/latex]<\/strong>has an initial point[latex]\\,{P}_{1}\\,[\/latex]and a terminal point[latex]\\,{P}_{2}\\,[\/latex]and <strong>[latex]v[\/latex]<\/strong> has an initial point[latex]\\,{P}_{3}\\,[\/latex]and a terminal point[latex]\\,{P}_{4}[\/latex].<\/p>\n\n<div id=\"fs-id1165133354243\">\n<div id=\"fs-id1165133354244\">\n<p id=\"fs-id1165133354245\">[latex]{P}_{1}=\\left(5,1\\right),{P}_{2}=\\left(3,-2\\right),{P}_{3}=\\left(-1,3\\right),\\,[\/latex]and[latex]\\,{P}_{4}=\\left(9,-4\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133022986\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133022986\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133022986\"]\n<p id=\"fs-id1165133022988\">not equal<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133022992\">\n<div id=\"fs-id1165133022994\">\n<p id=\"fs-id1165135534902\">[latex]{P}_{1}=\\left(2,-3\\right),{P}_{2}=\\left(5,1\\right),{P}_{3}=\\left(6,-1\\right),\\,[\/latex]and[latex]\\,{P}_{4}=\\left(9,3\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134032413\">\n<div id=\"fs-id1165134032414\">\n<p id=\"fs-id1165134573200\">[latex]{P}_{1}=\\left(-1,-1\\right),{P}_{2}=\\left(-4,5\\right),{P}_{3}=\\left(-10,6\\right),\\,[\/latex]and[latex]\\,{P}_{4}=\\left(-13,12\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137832972\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137832972\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137832972\"]\n<p id=\"fs-id1165137832974\">equal<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832978\">\n<div id=\"fs-id1165137832980\">\n<p id=\"fs-id1165137832981\">[latex]{P}_{1}=\\left(3,7\\right),{P}_{2}=\\left(2,1\\right),{P}_{3}=\\left(1,2\\right),\\,[\/latex]and[latex]\\,{P}_{4}=\\left(-1,-4\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133355961\">\n<div id=\"fs-id1165133355962\">\n<p id=\"fs-id1165133355963\">[latex]{P}_{1}=\\left(8,3\\right),{P}_{2}=\\left(6,5\\right),{P}_{3}=\\left(11,8\\right),\\,[\/latex]and[latex]{P}_{4}=\\left(9,10\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133341007\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133341007\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133341007\"]\n<p id=\"fs-id1165133341010\">equal<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133341014\">\n<div id=\"fs-id1165133341015\">\n<p id=\"fs-id1165133341016\">Given initial point[latex]\\,{P}_{1}=\\left(-3,1\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(5,2\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135464840\">\n<div id=\"fs-id1165135464841\">\n<p id=\"fs-id1165135464842\">Given initial point[latex]\\,{P}_{1}=\\left(6,0\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-1,-3\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165135662547\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135662547\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135662547\"]\n<p id=\"fs-id1165135662549\">[latex]7i-3j[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135499649\">For the following exercises, use the vectors <strong><em>u<\/em><\/strong> = <strong><em>i<\/em><\/strong> + 5<strong><em>j<\/em><\/strong>, <strong><em>v<\/em><\/strong> = \u22122<strong><em>i<\/em><\/strong>\u2212 3<strong><em>j<\/em><\/strong>,\u2009 and <strong><em>w<\/em><\/strong> = 4<strong><em>i<\/em><\/strong> \u2212 <strong><em>j<\/em><\/strong>.<\/p>\n\n<div id=\"fs-id1165135152214\">\n<div id=\"fs-id1165135152215\">\n<p id=\"fs-id1165135152216\">Find <strong><em>u<\/em><\/strong> + (<strong><em>v<\/em><\/strong> \u2212 <strong><em>w<\/em><\/strong>)<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135400265\">\n<div id=\"fs-id1165135400266\">\n<p id=\"fs-id1165135400267\">Find 4<strong><em>v<\/em><\/strong> + 2<strong><em>u<\/em><\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165133233962\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133233962\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133233962\"]\n<p id=\"fs-id1165135190041\">[latex]-6i-2j[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134247986\">For the following exercises, use the given vectors to compute <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>, and 2<strong><em>u<\/em><\/strong> \u2212 3<strong><em>v<\/em><\/strong>.<\/p>\n\n<div id=\"fs-id1165134437218\">\n<div id=\"fs-id1165134437219\">\n<p id=\"fs-id1165134437220\">[latex]u=\u23292,-3\u232a,v=\u23291,5\u232a[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137900278\">\n<div id=\"fs-id1165137900279\">\n<p id=\"fs-id1165137900280\">[latex]u=\u2329-3,4\u232a,v=\u2329-2,1\u232a[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137642613\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137642613\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137642613\"]\n<p id=\"fs-id1165137642615\">[latex]u+v=\u3008-5,5\u3009,u-v=\u3008-1,3\u3009,2u-3v=\u30080,5\u3009[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133300700\">\n<div id=\"fs-id1165133300701\">\n<p id=\"fs-id1165133300702\">Let <strong><em>v<\/em><\/strong> = \u22124<strong><em>i<\/em><\/strong> + 3<strong><em>j<\/em><\/strong>. Find a vector that is half the length and points in the same direction as<strong>[latex]\\,v.[\/latex]<\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134401632\">\n<div id=\"fs-id1165134401634\">\n<p id=\"fs-id1165134401635\">Let <strong><em>v<\/em><\/strong> = 5<strong><em>i<\/em><\/strong> + 2<strong><em>j<\/em><\/strong>. Find a vector that is twice the length and points in the opposite direction as<strong>[latex]\\,v.[\/latex]<\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165133366189\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133366189\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133366189\"]\n<p id=\"fs-id1165133366191\">[latex]-10i\u20134j[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134173717\">For the following exercises, find a unit vector in the same direction as the given vector.<\/p>\n\n<div id=\"fs-id1165137643853\">\n<div id=\"fs-id1165137643854\">\n<p id=\"fs-id1165137643855\"><strong><em>a<\/em><\/strong> = 3<strong><em>i<\/em><\/strong> + 4<strong><em>j<\/em><\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137883771\">\n<div>\n\n<strong><em>b<\/em><\/strong> = \u22122<strong><em>i<\/em><\/strong> + 5<strong><em>j<\/em><\/strong>\n\n<\/div>\n<div id=\"fs-id1165137892399\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137892399\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137892399\"]\n<p id=\"fs-id1165137892401\">[latex]-\\frac{2\\sqrt{29}}{29}i+\\frac{5\\sqrt{29}}{29}j[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134129835\">\n<p id=\"fs-id1165134129836\"><strong><em>c<\/em><\/strong> = 10<strong><em>i<\/em><\/strong> \u2013 <strong><em>j<\/em><\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134381775\">\n<div id=\"fs-id1165134381776\">\n<p id=\"fs-id1165134381777\">[latex]d=-\\frac{1}{3}i+\\frac{5}{2}j[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133408795\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133408795\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133408795\"]\n[latex]-\\frac{2\\sqrt{229}}{229}i+\\frac{15\\sqrt{229}}{229}j[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165134057517\">\n<div id=\"fs-id1165134057518\">\n<p id=\"fs-id1165134057519\"><strong><em>u<\/em><\/strong> = 100<strong><em>i<\/em><\/strong> + 200<strong><em>j<\/em><\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137846165\">\n<div id=\"fs-id1165137846166\">\n<p id=\"fs-id1165137846167\"><strong><em>u<\/em><\/strong> = \u221214<strong><em>i<\/em><\/strong> + 2<strong><em>j<\/em><\/strong><\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"546910\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"546910\"][latex]-\\frac{7\\sqrt{2}}{10}i+\\frac{\\sqrt{2}}{10}j[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135445726\">For the following exercises, find the magnitude and direction of the vector,[latex]\\,0\\le \\theta &lt;2\\pi .[\/latex]<\/p>\n\n<div id=\"fs-id1165132035955\">\n<div id=\"fs-id1165132035956\">\n<p id=\"fs-id1165132035957\">[latex]\u23290,4\u232a[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134113808\">\n<div id=\"fs-id1165134113810\">\n<p id=\"fs-id1165134113811\">[latex]\u23296,5\u232a[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137894302\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137894302\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137894302\"]\n<p id=\"fs-id1165137894304\">[latex]|v|=7.810,\\theta =39.806\u00b0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134544880\">\n<div id=\"fs-id1165137810093\">\n<p id=\"fs-id1165137810094\">[latex]\u23292,-5\u232a[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137835752\">\n<div id=\"fs-id1165137835753\">\n<p id=\"fs-id1165137835754\">[latex]\u2329-4,-6\u232a[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"741575\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"741575\"][latex]|v|=7.211,\\theta =236.310\u00b0[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134351886\">\n<div id=\"fs-id1165134351887\">\n<p id=\"fs-id1165134351888\">Given <strong><em>u<\/em><\/strong> = 3<strong><em>i<\/em><\/strong> \u2212 4<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = \u22122<strong><em>i<\/em><\/strong> + 3<strong><em>j<\/em><\/strong>, calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134540065\">\n<div id=\"fs-id1165134540066\">\n<p id=\"fs-id1165134540067\">Given <strong><em>u<\/em><\/strong> = \u2212<strong><em>i<\/em><\/strong> \u2212 <strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = <strong><em>i<\/em><\/strong> + 5<strong><em>j<\/em><\/strong>, calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165134185467\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134185467\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134185467\"]\n<p id=\"fs-id1165134185469\">[latex]-6[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134129815\">\n<div id=\"fs-id1165134129816\">\n<p id=\"fs-id1165134129817\">Given[latex]\\,u=\u2329-2,4\u232a\\,[\/latex]and[latex]\\,v=\u2329-3,1\u232a,\\,[\/latex]calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134192874\">\n<div id=\"fs-id1165134192875\">\n\nGiven <strong><em>u<\/em><\/strong>[latex]=\u2329-1,6\u232a[\/latex]and <strong><em>v<\/em><\/strong>[latex]=\u23296,-1\u232a,[\/latex]calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong>\n\n<\/div>\n<div id=\"fs-id1165131949016\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131949016\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165131949016\"]\n<p id=\"fs-id1165131949018\">[latex]-12[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133300653\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165134254430\">For the following exercises, given<strong>[latex]\\,v,\\,[\/latex]<\/strong>draw<strong>[latex]v,[\/latex]<\/strong>3<strong><em>v<\/em><\/strong> and[latex]\\,\\frac{1}{2}v.[\/latex]<\/p>\n\n<div id=\"fs-id1165135404720\">\n<div id=\"fs-id1165135404721\">\n<p id=\"fs-id1165135404722\">[latex]\u23292,-1\u232a[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135412891\">\n<div>[latex]\u2329-1,4\u232a[\/latex]<\/div>\n<div id=\"fs-id1165134554319\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134554319\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134554319\"]<span id=\"fs-id1165134554325\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153133\/CNX_Precalc_Figure_08_08_253.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135328732\">\n<div id=\"fs-id1165135328733\">\n<p id=\"fs-id1165135328734\">[latex]\u2329-3,-2\u232a[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135360308\">For the following exercises, use the vectors shown to sketch <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>, and 2<strong><em>u<\/em><\/strong>.<\/p>\n\n<div id=\"fs-id1165135347501\">\n<div id=\"fs-id1165135347502\"><span id=\"fs-id1165135347509\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153135\/CNX_Precalc_Figure_08_08_204.jpg\" alt=\"Plot of vectors u and v extending from the same origin point. In terms of that point, u goes to (1,1) and v goes to (-1,2).\"><\/span><\/div>\n<div id=\"fs-id1165134130859\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134130859\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134130859\"]<span id=\"fs-id1165134130866\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153141\/CNX_Precalc_Figure_08_08_205.jpg\" alt=\"Plot of u+v, u-v, and 2u based on the above vectors. In relation to the same origin point, u+v goes to (0,3), u-v goes to (2,-1), and 2u goes to (2,2).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165133056304\">\n<div id=\"fs-id1165133056305\"><span id=\"fs-id1165133056311\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153143\/CNX_Precalc_Figure_08_08_206.jpg\" alt=\"Plot of vectors u and v extending from the same origin point. In terms of that point, u goes to (1,2) and v goes to (1,-1).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135203268\">\n<div id=\"fs-id1165135203269\"><span id=\"fs-id1165135203274\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153149\/CNX_Precalc_Figure_08_08_208.jpg\" alt=\"Plot of vectors u and v located head to tail. Take u's start point as the origin. In terms of that, u goes from the origin to (3,-2), and v goes from (3,-2) to (2,-3)\"><\/span><\/div>\n<div id=\"fs-id1165135203284\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135203284\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135203284\"]<span id=\"fs-id1165135191577\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153151\/CNX_Precalc_Figure_08_08_209.jpg\" alt=\"Plot of vectors u+v, u-v, and 2u based on the above vectors.Given that u's start point was the origin, u+v starts at the origin and goes to (2,-3); u-v starts at the origin and goes to (4,-1); 2u goes from the origin to (6,-4).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1165135191588\">For the following exercises, use the vectors shown to sketch 2<strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>.<\/p>\n\n<div id=\"fs-id1165135347138\">\n<div id=\"fs-id1165135347139\"><span id=\"fs-id1165135347146\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153159\/CNX_Precalc_Figure_08_08_210.jpg\" alt=\"Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (3,1) and v goes from the origin to (2,-2).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134272782\">\n<div id=\"fs-id1165134272783\"><span id=\"fs-id1165134272790\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153202\/CNX_Precalc_Figure_08_08_212.jpg\" alt=\"Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (1,-2) and v goes from the origin to (-3,-2).\"><\/span><\/div>\n<div id=\"fs-id1165137696690\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137696690\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137696690\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153204\/CNX_Precalc_Figure_08_08_213.jpg\" alt=\"Plot of a single vector. Taking the start point of the vector as (0,0) from the above set up, the vector goes from the origin to (-1,-6).\">[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1165135509130\">For the following exercises, use the vectors shown to sketch <strong><em>u<\/em><\/strong> \u2212 3<strong><em>v<\/em><\/strong>.<\/p>\n\n<div id=\"fs-id1165134193570\">\n<div id=\"fs-id1165134193571\"><span id=\"fs-id1165133281363\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153207\/CNX_Precalc_Figure_08_08_214.jpg\" alt=\"Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (-4,0) and v goes from the origin to (1,-1).\"><\/span><\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134311986\"><span id=\"fs-id1165134311993\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153213\/CNX_Precalc_Figure_08_08_216.jpg\" alt=\"Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (1,2) and v goes from the origin to (-2,1).\"><\/span><\/div>\n<div id=\"fs-id1165135665485\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135665485\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135665485\"]<span id=\"fs-id1165135665494\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153216\/CNX_Precalc_Figure_08_08_217.jpg\" alt=\"Vector extending from the origin to (7,5), taking the base as the origin.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1165133266615\">For the following exercises, write the vector shown in component form.<\/p>\n\n<div id=\"fs-id1165133266619\">\n<div id=\"fs-id1165133266620\"><span id=\"fs-id1165134467613\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153226\/CNX_Precalc_Figure_08_08_218.jpg\" alt=\"Vector going from the origin to (-4,2).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134467625\">\n<div id=\"fs-id1165134467626\"><span id=\"fs-id1165133354213\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153235\/CNX_Precalc_Figure_08_08_219.jpg\" alt=\"Insert figure(table) alt text: Vector going from the origin to (4,1).\"><\/span><\/div>\n<div id=\"fs-id1165133354225\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133354225\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133354225\"]\n<p id=\"fs-id1165134267740\">[latex]\u30084,1\u3009[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135649444\">\n<div id=\"fs-id1165135649445\">\n<p id=\"fs-id1165135649446\">Given initial point[latex]\\,{P}_{1}=\\left(2,1\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-1,2\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j,\\,[\/latex]<\/strong>then draw the vector on the graph.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133213855\">\n<div id=\"fs-id1165133213856\">\n<p id=\"fs-id1165133213857\">Given initial point[latex]\\,{P}_{1}=\\left(4,-1\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-3,2\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong>Draw the points and the vector on the graph.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134179596\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134179596\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134179596\"]\n<p id=\"fs-id1165134179598\">[latex]v=-7i+3j[\/latex]<\/p>\n<span id=\"fs-id1165137892136\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153237\/CNX_Precalc_Figure_08_08_221.jpg\" alt=\"Vector going from (4,-1) to (-3,2).\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656840\">\n<div id=\"fs-id1165137656843\">\n<p id=\"fs-id1165137656845\">Given initial point[latex]\\,{P}_{1}=\\left(3,3\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-3,3\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong>Draw the points and the vector on the graph.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135247487\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165135247492\">For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.<\/p>\n\n<div id=\"fs-id1165135247497\">\n<div id=\"fs-id1165135247498\">\n<p id=\"fs-id1165134043940\">[latex]|v|=6,\\theta =45\u00b0[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137942347\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137942347\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137942347\"]\n<p id=\"fs-id1165137942349\">[latex]3\\sqrt{2}i+3\\sqrt{2}j[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135359683\">\n<div id=\"fs-id1165135359684\">\n<p id=\"fs-id1165135359685\">[latex]|v|=8,\\theta =220\u00b0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135181781\">\n<p id=\"fs-id1165135181782\">[latex]|v|=2,\\theta =300\u00b0[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137936571\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137936571\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137936571\"]\n<p id=\"fs-id1165137936573\">[latex]i-\\sqrt{3}j[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134394479\">\n<div id=\"fs-id1165134394480\">\n<p id=\"fs-id1165134394482\">[latex]|v|=5,\\theta =135\u00b0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661647\">\n<div id=\"fs-id1165137661648\">\n<p id=\"fs-id1165137661649\">A 60-pound box is resting on a ramp that is inclined 12\u00b0. Rounding to the nearest tenth,<\/p>\n\n<ol id=\"fs-id1165135397073\" type=\"a\">\n \t<li>Find the magnitude of the normal (perpendicular) component of the force.<\/li>\n \t<li>Find the magnitude of the component of the force that is parallel to the ramp.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135397085\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135397085\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135397085\"]\n<p id=\"fs-id1165135397087\">a. 58.7; b. 12.5<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135397090\">\n<div id=\"fs-id1165135397092\">\n<p id=\"fs-id1165135361159\">A 25-pound box is resting on a ramp that is inclined 8\u00b0. Rounding to the nearest tenth,<\/p>\n\n<ol id=\"fs-id1165135361164\" type=\"a\">\n \t<li>Find the magnitude of the normal (perpendicular) component of the force.<\/li>\n \t<li>Find the magnitude of the component of the force that is parallel to the ramp.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135361176\">\n<div id=\"fs-id1165134230559\">\n<p id=\"fs-id1165134230560\">Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of 27\u00b0 above the horizontal. Round to the nearest hundredth.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134230567\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134230567\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134230567\"]\n<p id=\"fs-id1165134230569\">[latex]x=7.13\\,[\/latex]pounds,[latex]\\,y=3.63\\,[\/latex]pounds<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135320315\">\n<div id=\"fs-id1165135320316\">\n<p id=\"fs-id1165135320317\">Find the magnitude of the horizontal and vertical components of the vector with magnitude 4 pounds pointed in a direction of 127\u00b0 above the horizontal. Round to the nearest hundredth.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135320323\">\n<div id=\"fs-id1165135320324\">\n<p id=\"fs-id1165135320325\">Find the magnitude of the horizontal and vertical components of a vector with magnitude 5 pounds pointed in a direction of 55\u00b0 above the horizontal. Round to the nearest hundredth.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134042375\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134042375\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134042375\"]\n<p id=\"fs-id1165134042378\">[latex]x=2.87\\,[\/latex]pounds,[latex]\\,y=4.10\\,[\/latex]pounds<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135452097\">\n<div id=\"fs-id1165135452098\">\n<p id=\"fs-id1165135452099\">Find the magnitude of the horizontal and vertical components of the vector with magnitude 1 pound pointed in a direction of 8\u00b0 above the horizontal. Round to the nearest hundredth.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135452106\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135452111\">\n<div id=\"fs-id1165134031246\">\n\nA woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?\n\n<\/div>\n<div id=\"fs-id1165134031254\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134031254\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134031254\"]\n<p id=\"fs-id1165134031256\">4.635 miles, 17.764\u00b0 N of E<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134031260\">\n<div id=\"fs-id1165134031261\">\n<p id=\"fs-id1165134031262\">A boat leaves the marina and sails 6 miles north, then 2 miles northeast. How far from the marina is the boat, and in what direction must it sail to head directly back to the marina?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135354908\">\n<div id=\"fs-id1165135354909\">\n<p id=\"fs-id1165135354910\">A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135354915\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135354915\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135354915\"]\n<p id=\"fs-id1165135354917\">17 miles. 10.318 miles<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135354921\">\n<div id=\"fs-id1165135354922\">\n<p id=\"fs-id1165134248817\">A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight home, how far would she have to walk?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137673347\">\n<div id=\"fs-id1165137673348\">\n<p id=\"fs-id1165137673349\">A man starts walking from home and walks 3 miles at 20\u00b0 north of west, then 5 miles at 10\u00b0 west of south, then 4 miles at 15\u00b0 north of east. If he walked straight home, how far would he have to the walk, and in what direction?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137673356\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137673356\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137673356\"]\n<p id=\"fs-id1165137673359\">Distance: 2.868. Direction: 86.474\u00b0 North of West, or 3.526\u00b0 West of North<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134116933\">\n<div id=\"fs-id1165134116934\">\n<p id=\"fs-id1165134116935\">A woman starts walking from home and walks 6 miles at 40\u00b0 north of east, then 2 miles at 15\u00b0 east of south, then 5 miles at 30\u00b0 south of west. If she walked straight home, how far would she have to walk, and in what direction?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134116942\">\n<div id=\"fs-id1165134116943\">\n<p id=\"fs-id1165134116944\">An airplane is heading north at an airspeed of 600 km\/hr, but there is a wind blowing from the southwest at 80 km\/hr. How many degrees off course will the plane end up flying, and what is the plane\u2019s speed relative to the ground?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134370072\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134370072\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134370072\"]\n<p id=\"fs-id1165134370074\">4.924\u00b0. 659 km\/hr<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134370078\">\n<div id=\"fs-id1165134370080\">\n\nAn airplane is heading north at an airspeed of 500 km\/hr, but there is a wind blowing from the northwest at 50 km\/hr. How many degrees off course will the plane end up flying, and what is the plane\u2019s speed relative to the ground?\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134040498\">\n<div id=\"fs-id1165134040499\">\n<p id=\"fs-id1165134040500\">An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km\/hr. The plane flies with an airspeed of 550 km\/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134040504\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134040504\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134040504\"]\n<p id=\"fs-id1165134040507\">4.424\u00b0<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134040511\">\n<div id=\"fs-id1165134040512\">\n<p id=\"fs-id1165134040513\">An airplane needs to head due north, but there is a wind blowing from the northwest at 80 km\/hr. The plane flies with an airspeed of 500 km\/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133306917\">\n<div id=\"fs-id1165133306918\">\n\nAs part of a video game, the point[latex]\\,\\left(5,7\\right)\\,[\/latex]is rotated counterclockwise about the origin through an angle of 35\u00b0. Find the new coordinates of this point.\n\n<\/div>\n<div id=\"fs-id1165135582005\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135582005\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135582005\"]\n<p id=\"fs-id1165134395208\">[latex]\\left(0.081,8.602\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134072229\">\n<div id=\"fs-id1165134072230\">\n<p id=\"fs-id1165134072231\">As part of a video game, the point[latex]\\,\\left(7,3\\right)\\,[\/latex]is rotated counterclockwise about the origin through an angle of 40\u00b0. Find the new coordinates of this point.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134398682\">\n<div id=\"fs-id1165134398683\">\n<p id=\"fs-id1165134398684\">Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134398690\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134398690\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134398690\"]\n<p id=\"fs-id1165134398693\">21.801\u00b0, relative to the car\u2019s forward direction<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134398697\">\n<div id=\"fs-id1165134255038\">\n<p id=\"fs-id1165134183764\">Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 8 mph relative to the car, and the car is traveling 45 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134183770\">\n<div id=\"fs-id1165134183771\">\n<p id=\"fs-id1165134183772\">A 50-pound object rests on a ramp that is inclined 19\u00b0. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134183778\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134183778\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134183778\"]\n<p id=\"fs-id1165134183781\">parallel: 16.28, perpendicular: 47.28 pounds<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279250\">\n<div id=\"fs-id1165134279251\">\n<p id=\"fs-id1165134279252\">Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it upward, and 5 pounds acting on it 45\u00b0 from the horizontal. What single force is the resultant force acting on the body?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279259\">\n<div id=\"fs-id1165134279260\">\n<p id=\"fs-id1165134279261\">Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it \u2500135\u00b0 from the horizontal, and 5 pounds acting on it directed 150\u00b0 from the horizontal. What single force is the resultant force acting on the body?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137920642\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137920642\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137920642\"]\n<p id=\"fs-id1165137920644\">19.35 pounds, 231.54\u00b0 from the horizontal<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137920648\">\n<div id=\"fs-id1165137920650\">\n<p id=\"fs-id1165137920651\">The condition of equilibrium is when the sum of the forces acting on a body is the zero vector. Suppose a body has a force of 2 pounds acting on it to the right, 5 pounds acting on it upward, and 3 pounds acting on it 45\u00b0 from the horizontal. What single force is needed to produce a state of equilibrium on the body?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137920659\">\n<div id=\"fs-id1165133309247\">\n<p id=\"fs-id1165135364108\">Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30\u00b0 from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.<\/p>\n\n<\/div>\n<div id=\"fs-id1165133309254\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133309254\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133309254\"]\n<p id=\"fs-id1165133309256\">5.1583 pounds, 75.8\u00b0 from the horizontal<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133309264\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1165134497721\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/aea1516b-dcb8-4853-a3b6-b15775067250\">Non-right Triangles: Law of Sines<\/a><\/h4>\n<p id=\"fs-id1165134497726\">For the following exercises, assume[latex]\\,\\alpha \\,[\/latex]is opposite side[latex]\\,a,\\beta \\,[\/latex]is opposite side[latex]\\,b,\\,[\/latex]and[latex]\\,\\gamma \\,[\/latex]is opposite side[latex]\\,c.\\,[\/latex]Solve each triangle, if possible. Round each answer to the nearest tenth.<\/p>\n\n<div id=\"fs-id1165135440338\">\n<div id=\"fs-id1165135440339\">\n<p id=\"fs-id1165135440340\">[latex]\\beta =50\u00b0,a=105,b=45[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134573204\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134573204\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134573204\"]\n<p id=\"fs-id1165134573206\">Not possible<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134573212\">\n<p id=\"fs-id1165134573213\">[latex]\\alpha =43.1\u00b0,a=184.2,b=242.8[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134170057\">\n<div id=\"fs-id1165134170058\">\n<p id=\"fs-id1165134170059\">Solve the triangle.<\/p>\n<span id=\"fs-id1165134036684\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153244\/CNX_Precalc_Figure_08_08_223.jpg\" alt=\"Triangle with standard labels. Angle A is 36 degrees with opposite side a unknown. Angle B is 24 degrees with opposite side b = 16. Angle C and side c are unknown.\"><\/span>\n\n<\/div>\n<div id=\"fs-id1165134036696\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134036696\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134036696\"]\n<p id=\"fs-id1165134036698\">[latex]C=120\u00b0,a=23.1,c=34.1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137936465\">\n<div id=\"fs-id1165137936466\">\n<p id=\"fs-id1165137936467\">Find the area of the triangle.<\/p>\n<span id=\"fs-id1165135317538\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153251\/CNX_Precalc_Figure_08_08_224.jpg\" alt=\"A triangle. One angle is 75 degrees with opposite side unknown. The adjacent sides to the 75 degree angle are 8 and 11.\"><\/span>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135317550\">\n<div id=\"fs-id1165135384884\">\n<p id=\"fs-id1165135384885\">A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25\u00b0 and 49\u00b0, as shown in <a class=\"autogenerated-content\" href=\"#Image_08_08_225\">(Figure)<\/a>. Find the distance of the plane from point[latex]\\,A\\,[\/latex]and the elevation of the plane.<\/p>\n\n<div id=\"Image_08_08_225\" class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153307\/CNX_Precalc_Figure_08_08_225.jpg\" alt=\"Diagram of a plane flying over a highway. It is to the left and above points A and B on the ground in that order. There is a horizontal line going through the plan parallel to the ground. The angle formed by the horizontal line, the plane, and the line from the plane to point B is 25 degrees. The angle formed by the horizontal line, the plane, and point A is 49 degrees.\" width=\"487\" height=\"201\"> <strong>Figure 20.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279590\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134279590\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134279590\"]\n<p id=\"fs-id1165134279592\">distance of the plane from point[latex]\\,A:\\,[\/latex]2.2 km, elevation of the plane: 1.6 km<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134122825\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f59bb6c3-d618-4ba3-a4ab-97bd701e6957\">Non-right Triangles: Law of Cosines<\/a><\/h4>\n<div id=\"fs-id1165133078109\">\n<div id=\"fs-id1165133078110\">\n<p id=\"fs-id1165133078111\">Solve the triangle, rounding to the nearest tenth, assuming[latex]\\,\\alpha \\,[\/latex]is opposite side[latex]\\,a,\\beta \\,[\/latex]is opposite side[latex]\\,b,\\,[\/latex]and[latex]\\,\\gamma \\,[\/latex]s opposite side[latex]c:\\,a=4, b=6,c=8.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137676087\">\n<div id=\"fs-id1165137676088\">\n<p id=\"fs-id1165137676089\">Solve the triangle in <a class=\"autogenerated-content\" href=\"#Image_08_08_226\">(Figure)<\/a>, rounding to the nearest tenth.<\/p>\n\n<div id=\"Image_08_08_226\" class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153310\/CNX_Precalc_Figure_08_08_226.jpg\" alt=\"A standardly labeled triangle. Angle A is 54 degrees with opposite side a unknown. Angle B is unknown with opposite side b=15. Angle C is unknown with opposite side C=13.\" width=\"487\" height=\"221\"> <strong>Figure 21.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134241045\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134241045\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134241045\"]\n<p id=\"fs-id1165134178492\">[latex]B=71.0\u00b0,C=55.0\u00b0,a=12.8[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135407410\">\n<div id=\"fs-id1165135407411\">\n<p id=\"fs-id1165131962194\">Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165131962197\">\n<div id=\"fs-id1165131962198\">\n<p id=\"fs-id1165131962199\">To find the distance between two cities, a satellite calculates the distances and angle shown in <a class=\"autogenerated-content\" href=\"#Image_08_08_227\">(Figure)<\/a> (not to scale). Find the distance between the cities. Round answers to the nearest tenth.<\/p>\n\n<div id=\"Image_08_08_227\" class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153312\/CNX_Precalc_Figure_08_08_227.jpg\" alt=\"Diagram of a satellite above and to the right of two cities. The distance from the satellite to the closer city is 210 km. The distance from the satellite to the further city is 250 km. The angle formed by the closer city, the satellite, and the other city is 1.8 degrees.\" width=\"488\" height=\"264\"> <strong>Figure 22.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135329641\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135329641\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135329641\"]\n<p id=\"fs-id1165135329643\">40.6 km<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134385712\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/c1e7a952-2a95-4202-b38b-126d1ea832fb\">Polar Coordinates<\/a><\/h4>\n<div id=\"fs-id1165134385717\">\n<div id=\"fs-id1165134385718\">\n<p id=\"fs-id1165134385719\">Plot the point with polar coordinates[latex]\\,\\left(3,\\frac{\\pi }{6}\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541519\">\n<div id=\"fs-id1165135541520\">\n<p id=\"fs-id1165135541521\">Plot the point with polar coordinates[latex]\\,\\left(5,-\\frac{2\\pi }{3}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134152554\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134152554\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134152554\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153315\/CNX_Precalc_Figure_08_08_229.jpg\" alt=\"Polar coordinate grid with a point plotted on the fifth concentric circle 2\/3 the way between pi and 3pi\/2 (closer to 3pi\/2).\">[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165132914075\">\n<div id=\"fs-id1165132914076\">\n<p id=\"fs-id1165132914077\">Convert[latex]\\,\\left(6,-\\frac{3\\pi }{4}\\right)\\,[\/latex]to rectangular coordinates.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137701664\">\n<div id=\"fs-id1165137701665\">\n<p id=\"fs-id1165137701666\">Convert[latex]\\,\\left(-2,\\frac{3\\pi }{2}\\right)\\,[\/latex]to rectangular coordinates.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135499576\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135499576\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135499576\"]\n<p id=\"fs-id1165135499579\">[latex]\\,\\left(0,2\\right)\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165131907335\">\n<div id=\"fs-id1165131907336\">\n<p id=\"fs-id1165131907337\">Convert[latex]\\left(7,-2\\right)[\/latex]to polar coordinates.<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135367681\">\n<p id=\"fs-id1165135367682\">Convert[latex]\\left(-9,-4\\right)[\/latex]\nto polar coordinates.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135440152\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135440152\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135440152\"]\n<p id=\"fs-id1165135440154\">[latex]\\left(9.8489,203.96\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135640531\">For the following exercises, convert the given Cartesian equation to a polar equation.<\/p>\n\n<div id=\"fs-id1165135640534\">\n<div id=\"fs-id1165135640535\">\n<p id=\"fs-id1165135640536\">[latex]x=-2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137760868\">\n<div id=\"fs-id1165137760869\">\n<p id=\"fs-id1165137760870\">[latex]{x}^{2}+{y}^{2}=64[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133001914\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133001914\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133001914\"]\n<p id=\"fs-id1165133001916\">[latex]r=8[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134190679\">\n<div>\n<p id=\"fs-id1165134190681\">[latex]{x}^{2}+{y}^{2}=-2y[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135388841\">For the following exercises, convert the given polar equation to a Cartesian equation.<\/p>\n\n<div id=\"fs-id1165135388844\">\n<div id=\"fs-id1165135388845\">\n<p id=\"fs-id1165135388846\">[latex]r=7\\text{cos}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134138581\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134138581\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134138581\"]\n<p id=\"fs-id1165134138583\">[latex]{x}^{2}+{y}^{2}=7x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135331692\">\n<div id=\"fs-id1165135331694\">\n<p id=\"fs-id1165135436286\">[latex]r=\\frac{-2}{4\\mathrm{cos}\\,\\theta +\\mathrm{sin}\\,\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165133078070\">For the following exercises, convert to rectangular form and graph.<\/p>\n\n<div id=\"fs-id1165133078073\">\n<div id=\"fs-id1165133078074\">\n<p id=\"fs-id1165133078075\">[latex]\\theta =\\frac{3\\pi }{4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135521226\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135521226\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135521226\"]\n<p id=\"fs-id1165135521228\">[latex]y=-x[\/latex]<\/p>\n<span id=\"fs-id1165137853290\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153324\/CNX_Precalc_Figure_08_08_230.jpg\" alt=\"Plot of the function y=-x in rectangular coordinates.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133437206\">\n<div id=\"fs-id1165133437207\">\n<p id=\"fs-id1165133437208\">[latex]r=5\\mathrm{sec}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/a63ed4d5-a31b-403b-9aaa-80329d8bcaa0\">Polar Coordinates: Graphs<\/a><\/h4>\nFor the following exercises, test each equation for symmetry.\n<div>\n<div id=\"fs-id1165135618275\">[latex]r=4+4\\mathrm{sin}\\,\\theta [\/latex]<\/div>\n<div id=\"fs-id1165135519207\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135519207\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135519207\"]\n<p id=\"fs-id1165135519209\">symmetric with respect to the line[latex]\\theta =\\frac{\\pi }{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135442491\">\n<div id=\"fs-id1165135442492\">\n<p id=\"fs-id1165135442493\">[latex]r=7[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134430440\">\n<p id=\"fs-id1165134430441\">Sketch a graph of the polar equation[latex]\\,r=1-5\\mathrm{sin}\\,\\theta .\\,[\/latex]Label the axis intercepts.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134043892\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134043892\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134043892\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153327\/CNX_Precalc_Figure_08_08_232.jpg\" alt=\"Graph of the given polar equation - an inner loop lima\u00e7on.\">[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135403257\">\n<div id=\"fs-id1165135403258\">\n<p id=\"fs-id1165135403259\">Sketch a graph of the polar equation[latex]\\,r=5\\mathrm{sin}\\left(7\\theta \\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137756884\">\n<div id=\"fs-id1165137756885\">\n<p id=\"fs-id1165137756886\">Sketch a graph of the polar equation[latex]\\,r=3-3\\mathrm{cos}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134534234\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134534234\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134534234\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153334\/CNX_Precalc_Figure_08_08_234.jpg\" alt=\"Graph of the given polar equation - a cardioid.\">[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135252219\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/5217ca36-7905-420a-b241-8d24df993e56\">Polar Form of Complex Numbers<\/a><\/h4>\n<p id=\"fs-id1165135252224\">For the following exercises, find the absolute value of each complex number.<\/p>\n\n<div id=\"fs-id1165135252227\">\n<div id=\"fs-id1165135252228\">[latex]-2+6i[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135209072\">\n<div id=\"fs-id1165135209073\">\n<p id=\"fs-id1165135209074\">[latex]4-\\text{\u200b}3i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135337142\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135337142\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135337142\"]\n<p id=\"fs-id1165135337144\">5<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135337148\">Write the complex number in polar form.<\/p>\n\n<div id=\"fs-id1165135337152\">\n<div id=\"fs-id1165135337153\">\n<p id=\"fs-id1165135337154\">[latex]5+9i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135332336\">\n<div id=\"fs-id1165135332337\">\n<p id=\"fs-id1165135332338\">[latex]\\frac{1}{2}-\\frac{\\sqrt{3}}{2}\\text{\u200b}i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137707288\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137707288\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137707288\"]\n<p id=\"fs-id1165137707290\">[latex]\\mathrm{cis}\\left(-\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165133023588\">For the following exercises, convert the complex number from polar to rectangular form.<\/p>\n\n<div id=\"fs-id1165133023591\">\n<div id=\"fs-id1165133023592\">\n<p id=\"fs-id1165134385585\">[latex]z=5\\mathrm{cis}\\left(\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134205830\">\n<div id=\"fs-id1165134205831\">\n<p id=\"fs-id1165134205832\">[latex]z=3\\mathrm{cis}\\left(40\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133318597\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133318597\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133318597\"]\n<p id=\"fs-id1165133318599\">[latex]2.3+1.9i[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135360321\">For the following exercises, find the product[latex]\\,{z}_{1}{z}_{2}\\,[\/latex]in polar form.<\/p>\n\n<div id=\"fs-id1165135319994\">\n<div id=\"fs-id1165135319995\">\n\n[latex]{z}_{1}=2\\mathrm{cis}\\left(89\u00b0\\right)[\/latex]\n<p id=\"fs-id1165134159678\">[latex]{z}_{2}=5\\mathrm{cis}\\left(23\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134325188\">\n<div id=\"fs-id1165134325189\">\n<p id=\"fs-id1165134325191\">[latex]{z}_{1}=10\\mathrm{cis}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<p id=\"fs-id1165134313320\">[latex]{z}_{2}=6\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135702689\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135702689\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135702689\"]\n<p id=\"fs-id1165135702692\">[latex]60\\mathrm{cis}\\left(\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165131883365\">For the following exercises, find the quotient[latex]\\,\\frac{{z}_{1}}{{z}_{2}}\\,[\/latex]in polar form.<\/p>\n\n<div id=\"fs-id1165135255380\">\n<div id=\"fs-id1165135255382\">\n\n[latex]{z}_{1}=12\\mathrm{cis}\\left(55\u00b0\\right)[\/latex]\n<p id=\"fs-id1165133221828\">[latex]{z}_{2}=3\\mathrm{cis}\\left(18\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133178324\">\n<div id=\"fs-id1165133178326\">\n<p id=\"fs-id1165133178328\">[latex]{z}_{1}=27\\mathrm{cis}\\left(\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n<p id=\"fs-id1165131968026\">[latex]{z}_{2}=9\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165131863139\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131863139\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165131863139\"]\n<p id=\"fs-id1165131863141\">[latex]3\\mathrm{cis}\\left(\\frac{4\\pi }{3}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134216196\">For the following exercises, find the powers of each complex number in polar form.<\/p>\n\n<div id=\"fs-id1165134216200\">\n<div id=\"fs-id1165134216201\">\n<p id=\"fs-id1165135332898\">Find[latex]\\,{z}^{4}\\,[\/latex]when[latex]\\,z=2\\mathrm{cis}\\left(70\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132059611\">\n<div id=\"fs-id1165132059612\">\n<p id=\"fs-id1165132059613\">Find[latex]\\,{z}^{2}\\,[\/latex]when[latex]\\,z=5\\mathrm{cis}\\left(\\frac{3\\pi }{4}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133155816\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133155816\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133155816\"]\n<p id=\"fs-id1165133155818\">[latex]25\\mathrm{cis}\\left(\\frac{3\\pi }{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135694330\">For the following exercises, evaluate each root.<\/p>\n\n<div id=\"fs-id1165135694334\">\n<div id=\"fs-id1165135694335\">\n<p id=\"fs-id1165135694336\">Evaluate the cube root of[latex]\\,z\\,[\/latex]when[latex]\\,z=64\\mathrm{cis}\\left(210\u00b0\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133103838\">\n<div id=\"fs-id1165133103839\">\n<p id=\"fs-id1165133103840\">Evaluate the square root of[latex]\\,z\\,[\/latex]when[latex]\\,z=25\\mathrm{cis}\\left(\\frac{3\\pi }{2}\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135701699\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135701699\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135701699\"]\n<p id=\"fs-id1165135701701\">[latex]5\\mathrm{cis}\\left(\\frac{3\\pi }{4}\\right),5\\mathrm{cis}\\left(\\frac{7\\pi }{4}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137847075\">For the following exercises, plot the complex number in the complex plane.<\/p>\n\n<div id=\"fs-id1165137847078\">\n<div id=\"fs-id1165137847080\">\n<p id=\"fs-id1165137847081\">[latex]6-2i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132968119\">\n<div id=\"fs-id1165132968120\">\n<p id=\"fs-id1165132968121\">[latex]-1+3i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137937062\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137937062\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137937062\"]<span id=\"fs-id1165137937070\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153337\/CNX_Precalc_Figure_08_08_236n.jpg\" alt=\"Plot of -1 + 3i in the complex plane (-1 along the real axis, 3 along the imaginary).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133237118\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/908896ca-db26-40f9-99de-79070485b4e7\">Parametric Equations<\/a><\/h4>\n<p id=\"fs-id1165133237123\">For the following exercises, eliminate the parameter[latex]\\,t\\,[\/latex]to rewrite the parametric equation as a Cartesian equation.<\/p>\n\n<div id=\"fs-id1165133349424\">\n<div>\n<p id=\"fs-id1165133349426\">[latex]\\{\\begin{array}{l}x(t)=3t-1\\hfill \\\\ y(t)=\\sqrt{t}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134234217\">\n<div id=\"fs-id1165134234218\">\n<p id=\"fs-id1165134234219\">[latex]\\{\\begin{array}{l}x(t)=-\\mathrm{cos}\\,t\\hfill \\\\ y(t)=2{\\mathrm{sin}}^{2}t \\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135407436\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135407436\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135407436\"]\n<p id=\"fs-id1165135407438\">[latex]{x}^{2}+\\frac{1}{2}y=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135321170\">\n<div id=\"fs-id1165135321171\">\n<p id=\"fs-id1165135321172\">Parameterize (write a parametric equation for) each Cartesian equation by using[latex]\\,x\\left(t\\right)=a\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,y\\left(t\\right)=b\\mathrm{sin}\\,t\\,[\/latex]for[latex]\\,\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{16}=1.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135512804\">\n<div id=\"fs-id1165135512805\">\n<p id=\"fs-id1165135512806\">Parameterize the line from[latex]\\,\\left(-2,3\\right)\\,[\/latex]to[latex]\\,\\left(4,7\\right)\\,[\/latex]so that the line is at[latex]\\,\\left(-2,3\\right)\\,[\/latex]at[latex]\\,t=0\\,[\/latex]and[latex]\\,\\left(4,7\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134177538\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134177538\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134177538\"]\n<p id=\"fs-id1165134177540\">[latex]\\{\\begin{array}{l}x(t)=-2+6t\\hfill \\\\ y(t)=3+4t\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/38d919ed-1bee-4ae6-aed0-48285d870dab\">Parametric Equations: Graphs<\/a><\/h4>\n<p id=\"fs-id1165134177551\">For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.<\/p>\n\n<div id=\"fs-id1165134495161\">\n<div id=\"fs-id1165134495162\">\n<p id=\"fs-id1165134495163\">[latex]\\{\\begin{array}{l}x(t)=3{t}^{2}\\hfill \\\\ y(t)=2t-1\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135455023\">\n<div id=\"fs-id1165135455024\">\n<p id=\"fs-id1165135455026\">[latex]\\{\\begin{array}{l}x(t)={e}^{t}\\hfill \\\\ y(t)=-2{e}^{5\\,t}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134272725\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134272725\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134272725\"]\n<p id=\"fs-id1165135609213\">[latex]y=-2{x}^{5}[\/latex]<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153340\/CNX_Precalc_Figure_08_08_238.jpg\" alt=\"Plot of the given parametric equations.\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134185411\">\n<div id=\"fs-id1165134534274\">\n<p id=\"fs-id1165134534275\">[latex]\\{\\begin{array}{l}x(t)=3\\mathrm{cos}\\,t\\hfill \\\\ y(t)=2\\mathrm{sin}\\,t\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165131962229\">\n<p id=\"fs-id1165131962230\">A ball is launched with an initial velocity of 80 feet per second at an angle of 40\u00b0 to the horizontal. The ball is released at a height of 4 feet above the ground.<\/p>\n\n<ol id=\"fs-id1165131962236\" type=\"a\">\n \t<li>Find the parametric equations to model the path of the ball.<\/li>\n \t<li>Where is the ball after 3 seconds?<\/li>\n \t<li>How long is the ball in the air?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135472893\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135472893\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135472893\"]\n<ol id=\"fs-id1165135472895\" type=\"a\">\n \t<li>[latex]\\{\\begin{array}{l}x(t)=(80\\mathrm{cos}(40\u00b0))t\\\\ y(t)=-16{t}^{2}+(80\\mathrm{sin}(40\u00b0))t+4\\end{array}[\/latex]<\/li>\n \t<li>The ball is 14 feet high and 184 feet from where it was launched.<\/li>\n \t<li>3.3 seconds<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134356905\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/907cf72a-9d36-4043-98b4-3d251188ac6f\">Vectors<\/a><\/h4>\n<p id=\"fs-id1165135673959\">For the following exercises, determine whether the two vectors,<strong>[latex]\\,u\\,[\/latex]<\/strong>and<strong>[latex]\\,v,\\,[\/latex]<\/strong>are equal, where<strong>[latex]\\,u\\,[\/latex]<\/strong>has an initial point[latex]\\,{P}_{1}\\,[\/latex]and a terminal point[latex]\\,{P}_{2},\\,[\/latex]and<strong>[latex]\\,v\\,[\/latex]<\/strong>has an initial point[latex]\\,{P}_{3}\\,[\/latex]and a terminal point[latex]\\,{P}_{4}.[\/latex]<\/p>\n\n<div>\n<div>\n<p id=\"fs-id1165134534170\">[latex]{P}_{1}=\\left(-1,4\\right),{P}_{2}=\\left(3,1\\right),{P}_{3}=\\left(5,5\\right)[\/latex]and[latex]\\,{P}_{4}=\\left(9,2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137580687\">\n<div id=\"fs-id1165137580688\">\n<p id=\"fs-id1165137580690\">[latex]{P}_{1}=\\left(6,11\\right),{P}_{2}=\\left(-2,8\\right),{P}_{3}=\\left(0,-1\\right)\\,[\/latex]and[latex]\\,{P}_{4}=\\left(-8,2\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137780030\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137780030\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137780030\"]\n<p id=\"fs-id1165137780032\">not equal<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135532362\">For the following exercises, use the vectors<strong>[latex]\\,u=2i-j\\text{,}v=4i-3j\\text{,}\\,[\/latex]<\/strong>and<strong>[latex]\\,w=-2i+5j\\,[\/latex]<\/strong>to evaluate the expression.<\/p>\n\n<div id=\"fs-id1165137734400\">\n<div id=\"fs-id1165137734401\">\n<p id=\"fs-id1165137734402\"><strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165131886725\">\n<div id=\"fs-id1165131886726\">\n<p id=\"fs-id1165131886727\">2<strong><em>v<\/em><\/strong> \u2212 <strong><em>u<\/em><\/strong> + <strong><em>w<\/em><\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165133359377\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133359377\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133359377\"]\n<p id=\"fs-id1165133359379\">4<strong><em>i<\/em><\/strong><\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137895109\">For the following exercises, find a unit vector in the same direction as the given vector.<\/p>\n\n<div id=\"fs-id1165137895112\">\n<div id=\"fs-id1165137895113\">\n<p id=\"fs-id1165137895114\"><strong><em>a<\/em><\/strong> = 8<strong><em>i<\/em><\/strong> \u2212 6<strong><em>j<\/em><\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137938385\">\n<div id=\"fs-id1165137938386\">\n<p id=\"fs-id1165137938387\"><strong><em>b<\/em><\/strong> = \u22123<strong><em>i<\/em><\/strong> \u2212 <strong><em>j<\/em><\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165135351469\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135351469\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135351469\"]\n<p id=\"fs-id1165135351471\">[latex]-\\frac{3\\sqrt{10}}{10}[\/latex]<strong><em>i<\/em><\/strong>[latex]-\\frac{\\sqrt{10}}{10}[\/latex]<strong><em>j<\/em><\/strong><\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\nFor the following exercises, find the magnitude and direction of the vector.\n<div id=\"fs-id1165135252126\">\n<div>\n<p id=\"fs-id1165135252128\">[latex]\u23296,-2\u232a[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135189796\">\n<div id=\"fs-id1165135189797\">\n<p id=\"fs-id1165135189798\">[latex]\u2329-3,-3\u232a[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165132936410\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132936410\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165132936410\"]\n<p id=\"fs-id1165132936412\">Magnitude:[latex]\\,3\\sqrt{2},\\,[\/latex]Direction:[latex]\\text{225\u00b0}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165132934428\">For the following exercises, calculate<strong>[latex]\\,u\\cdot v\\text{.}[\/latex]<\/strong><\/p>\n\n<div id=\"fs-id1165134179615\">\n<div id=\"fs-id1165134179616\">\n<p id=\"fs-id1165134179618\"><strong><em>u<\/em><\/strong> = \u22122<strong><em>i<\/em><\/strong> + <strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = 3<strong><em>i<\/em><\/strong> + 7<strong><em>j<\/em><\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134248828\">\n<div id=\"fs-id1165134248829\">\n<p id=\"fs-id1165134248830\"><strong><em>u<\/em><\/strong> = <strong><em>i<\/em><\/strong> + 4<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = 4<strong><em>i<\/em><\/strong> + 3<strong><em>j<\/em><\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165137832740\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137832740\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137832740\"]\n<p id=\"fs-id1165137832743\">[latex]\\text{16}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832747\">\n<div id=\"fs-id1165137832748\">\n<p id=\"fs-id1165137832749\">Given <strong><em>v<\/em><\/strong>[latex]=\u3008-3,4\u3009[\/latex]draw <strong><em>v<\/em><\/strong>, 2<strong><em>v<\/em><\/strong>, and [latex]\\,\\frac{1}{2}[\/latex]<strong><em>v<\/em><\/strong>.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135694955\">\n<div id=\"fs-id1165135694956\">\n<p id=\"fs-id1165135694957\">Given the vectors shown in <a class=\"autogenerated-content\" href=\"#Image_08_08_241\">(Figure)<\/a>, sketch <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong> and 3<strong><em>v<\/em><\/strong>.<\/p>\n\n<div id=\"Image_08_08_241\" class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153352\/CNX_Precalc_Figure_08_08_241.jpg\" alt=\"Diagram of vectors v, 2v, and 1\/2 v. The 2v vector is in the same direction as v but has twice the magnitude. The 1\/2 v vector is in the same direction as v but has half the magnitude.\" width=\"487\" height=\"323\"> <strong>Figure 23.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135532583\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135532583\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135532583\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153354\/CNX_Precalc_Figure_08_08_242.jpg\" alt=\"Diagram of vectors u and v. Taking u's starting point as the origin, u goes from the origin to (4,1), and v goes from (4,1) to (6,0).\">[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165132957216\">\n<div id=\"fs-id1165132957217\">\n<p id=\"fs-id1165132957218\">Given initial point[latex]\\,{P}_{1}=\\left(3,2\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-5,-1\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,\\,i\\,\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong>Draw the points and the vector on the graph.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135179890\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<div id=\"fs-id1165135179893\">\n<div id=\"fs-id1165135179894\">\n<p id=\"fs-id1165135179895\">Assume[latex]\\,\\alpha \\,[\/latex]is opposite side[latex]\\,a,\\beta \\,[\/latex]is opposite side[latex]\\,b,\\,[\/latex]and[latex]\\,\\gamma \\,[\/latex]is opposite side[latex]\\,c.\\,[\/latex]Solve the triangle, if possible, and round each answer to the nearest tenth, given[latex]\\,\\beta =68\u00b0,b=21,c=16.[\/latex]<\/p>\n\n<\/div>\n<div>[latex]\\alpha =67.1\u00b0,\\gamma =44.9\u00b0,a=20.9[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134230590\">\n<div id=\"fs-id1165134230591\">\n<p id=\"fs-id1165134230592\">Find the area of the triangle in <a class=\"autogenerated-content\" href=\"#Image_08_08_244\">(Figure)<\/a>. Round each answer to the nearest tenth.<\/p>\n\n<div id=\"Image_08_08_244\" class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153357\/CNX_Precalc_Figure_08_08_244.jpg\" alt=\"A triangle. One angle is 60 degrees with opposite side 6.25. The other two sides are 5 and 7.\" width=\"487\" height=\"165\"> <strong>Figure 24.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134301539\">\n<div id=\"fs-id1165134301540\">\n<p id=\"fs-id1165134301541\">A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15\u00b0 to the right of his original course, and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134301549\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134301549\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134301549\"]\n<p id=\"fs-id1165134301551\">[latex]\\text{1712 miles}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134086133\">\n<p id=\"fs-id1165134086134\">Convert[latex]\\,\\left(2,2\\right)\\,[\/latex]\nto polar coordinates, and then plot the point.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133200949\">\n<div id=\"fs-id1165133200950\">\n<p id=\"fs-id1165133200951\">Convert[latex]\\,\\left(2,\\frac{\\pi }{3}\\right)\\,[\/latex]to rectangular coordinates.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135473730\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135473730\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135473730\"]\n<p id=\"fs-id1165135473732\">[latex]\\left(1,\\sqrt{3}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134301575\">\n<div id=\"fs-id1165134301576\">\n<p id=\"fs-id1165134301578\">Convert the polar equation to a Cartesian equation:[latex]\\,{x}^{2}+{y}^{2}=5\\mathrm{y.}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135459827\">\n\nConvert to rectangular form and graph:[latex]r=-3\\mathrm{csc}\\,\\theta .[\/latex]\n\n<\/div>\n<div id=\"fs-id1165133359307\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133359307\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133359307\"]\n<p id=\"fs-id1165133359309\">[latex]y=-3[\/latex]<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153359\/CNX_Precalc_Figure_08_08_246.jpg\" alt=\"Plot of the given equation in rectangular form - line y=-3.\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134085542\">\n<div id=\"fs-id1165134085543\">\n<p id=\"fs-id1165134085544\">Test the equation for symmetry:[latex]\\,r=-4\\mathrm{sin}\\left(2\\theta \\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133363596\">\n<div id=\"fs-id1165133363598\">\n<p id=\"fs-id1165133363599\">Graph[latex]\\,r=3+3\\mathrm{cos}\\,\\theta .[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134081559\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134081559\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134081559\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153407\/CNX_Precalc_Figure_08_08_247.jpg\" alt=\"Graph of the given equations - a cardioid.\">[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165134081170\">\n<div id=\"fs-id1165134081172\">\n<p id=\"fs-id1165134081174\">Graph[latex]\\,r=3-5\\text{sin}\\,\\theta .[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134532849\">\n<div id=\"fs-id1165134532851\">\n<p id=\"fs-id1165134532853\">Find the absolute value of the complex number\n[latex]5-9i.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165131841660\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131841660\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165131841660\"]\n<p id=\"fs-id1165131841662\">[latex]\\sqrt{106}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134038130\">\n<div id=\"fs-id1165134038131\">\n<p id=\"fs-id1165134038132\">Write the complex number in polar form:[latex]\\,4+i\\text{.}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165134070883\">Convert the complex number from polar to rectangular form:[latex]\\,z=5\\text{cis}\\left(\\frac{2\\pi }{3}\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137734329\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137734329\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137734329\"]\n<p id=\"fs-id1165137734331\">[latex]\\frac{-5}{2}+i\\frac{5\\sqrt{3}}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135426348\">Given[latex]\\,{z}_{1}=8\\mathrm{cis}\\left(36\u00b0\\right)\\,[\/latex]and[latex]\\,{z}_{2}=2\\mathrm{cis}\\left(15\u00b0\\right),[\/latex]evaluate each expression.<\/p>\n\n<div id=\"fs-id1165134036642\">\n<div id=\"fs-id1165134036644\">\n<p id=\"fs-id1165134036646\">[latex]{z}_{1}{z}_{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135662488\">\n<div id=\"fs-id1165135662490\">\n<p id=\"fs-id1165135662492\">[latex]\\frac{{z}_{1}}{{z}_{2}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135653907\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135653907\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135653907\"]\n<p id=\"fs-id1165135653909\">[latex]4\\mathrm{cis}\\left(21\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135305744\">\n<div id=\"fs-id1165135305746\">\n<p id=\"fs-id1165135305748\">[latex]{\\left({z}_{2}\\right)}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134430302\">\n<div id=\"fs-id1165134430304\">\n<p id=\"fs-id1165134430306\">[latex]\\sqrt{{z}_{1}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135537343\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135537343\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135537343\"]\n<p id=\"fs-id1165135537345\">[latex]2\\sqrt{2}\\mathrm{cis}\\left(18\u00b0\\right),2\\sqrt{2}\\mathrm{cis}\\left(198\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132955611\">\n<div id=\"fs-id1165132955612\">\n<p id=\"fs-id1165132955613\">Plot the complex number[latex]\\,-5-i\\,[\/latex]in the complex plane.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135432745\">\n<div id=\"fs-id1165135432746\">\n<p id=\"fs-id1165135432747\">Eliminate the parameter[latex]\\,t\\,[\/latex]to rewrite the following parametric equations as a Cartesian equation: [latex]\\,\\{\\begin{array}{l}x(t)=t+1\\hfill \\\\ y(t)=2{t}^{2}\\hfill \\end{array}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134179578\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134179578\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134179578\"]\n<p id=\"fs-id1165134179580\">[latex]y=2{\\left(x-1\\right)}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135342087\">\n<div id=\"fs-id1165135342089\">\n<p id=\"fs-id1165135342091\">Parameterize (write a parametric equation for) the following Cartesian equation by using[latex]\\,x\\left(t\\right)=a\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,y\\left(t\\right)=b\\mathrm{sin}\\,t:[\/latex][latex]\\frac{{x}^{2}}{36}+\\frac{{y}^{2}}{100}=1.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135359801\">\n<div id=\"fs-id1165135359803\">\n<p id=\"fs-id1165135359805\">Graph the set of parametric equations and find the Cartesian equation:[latex]\\,\\{\\begin{array}{l}x(t)=-2\\mathrm{sin}\\,t\\hfill \\\\ y(t)=5\\mathrm{cos}\\,t\\hfill \\end{array}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135636921\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135636921\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135636921\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153409\/CNX_Precalc_Figure_08_08_250.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\">[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165134053928\">\n<div id=\"fs-id1165137921477\">\n<p id=\"fs-id1165137921480\">A ball is launched with an initial velocity of 95 feet per second at an angle of 52\u00b0 to the horizontal. The ball is released at a height of 3.5 feet above the ground.<\/p>\n\n<ol id=\"fs-id1165137921485\" type=\"a\">\n \t<li>Find the parametric equations to model the path of the ball.<\/li>\n \t<li>Where is the ball after 2 seconds?<\/li>\n \t<li>How long is the ball in the air?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137921500\">For the following exercises, use the vectors <strong><em>u<\/em><\/strong> = <strong><em>i<\/em><\/strong> \u2212 3<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = 2<strong><em>i<\/em><\/strong> + 3<strong><em>j<\/em><\/strong>.<\/p>\n\n<div id=\"fs-id1165134257477\">\n<div id=\"fs-id1165134257480\">\n<p id=\"fs-id1165134257482\">Find 2<strong><em>u<\/em><\/strong> \u2212 3<strong><em>v<\/em><\/strong>.<\/p>\n\n<\/div>\n<div id=\"fs-id1165131835164\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131835164\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165131835164\"]\n<p id=\"fs-id1165131835167\">\u22124<strong><em>i<\/em><\/strong> \u2212 15<strong><em>j<\/em><\/strong><\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135673438\">\n<div id=\"fs-id1165135673439\">\n<p id=\"fs-id1165135673440\">Calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134060262\">\n<div id=\"fs-id1165134060263\">\n<p id=\"fs-id1165134060264\">Find a unit vector in the same direction as<strong>[latex]\\,v.[\/latex]<\/strong><\/p>\n\n<\/div>\n<div id=\"fs-id1165137772302\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137772302\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137772302\"]\n<p id=\"fs-id1165137772304\">[latex]\\frac{2\\sqrt{13}}{13}i+\\frac{3\\sqrt{13}}{13}j[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132039481\">\n<div id=\"fs-id1165132039482\">\n<p id=\"fs-id1165132039483\">Given vector<strong>[latex]\\,v\\,[\/latex]<\/strong>has an initial point[latex]\\,{P}_{1}=\\left(2,2\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-1,0\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong>On the graph, draw<strong>[latex]\\,v,\\,[\/latex]<\/strong>and<strong>[latex]\\,-v.\\,[\/latex]<\/strong><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165133447852\">\n \t<dt>dot product<\/dt>\n \t<dd id=\"fs-id1165133447857\">given two vectors, the sum of the product of the horizontal components and the product of the vertical components<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133447862\">\n \t<dt>initial point<\/dt>\n \t<dd id=\"fs-id1165135369492\">the origin of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135369497\">\n \t<dt>magnitude<\/dt>\n \t<dd id=\"fs-id1165135369502\">the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135369506\">\n \t<dt>resultant<\/dt>\n \t<dd id=\"fs-id1165135369512\">a vector that results from addition or subtraction of two vectors, or from scalar multiplication<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238406\">\n \t<dt>scalar<\/dt>\n \t<dd id=\"fs-id1165135238411\">a quantity associated with magnitude but not direction; a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238414\">\n \t<dt>scalar multiplication<\/dt>\n \t<dd id=\"fs-id1165135238419\">the product of a constant and each component of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238423\">\n \t<dt>standard position<\/dt>\n \t<dd id=\"fs-id1165135369538\">the placement of a vector with the initial point at[latex]\\,\\left(0,0\\right)\\,[\/latex]and the terminal point[latex]\\,\\left(a,b\\right),\\,[\/latex]represented by the change in the <em>x<\/em>-coordinates and the change in the <em>y<\/em>-coordinates of the original vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134036770\">\n \t<dt>terminal point<\/dt>\n \t<dd id=\"fs-id1165133243502\">the end point of a vector, usually represented by an arrow indicating its direction<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133243505\">\n \t<dt>unit vector<\/dt>\n \t<dd id=\"fs-id1165133243510\">a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the <em>x<\/em>-axis and is defined as[latex]\\,{v}_{1}=\u23291,0\u232a\\,[\/latex]the vertical unit vector runs along the <em>y<\/em>-axis and is defined as[latex]\\,{v}_{2}=\u23290,1\u232a.[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131906701\">\n \t<dt>vector<\/dt>\n \t<dd id=\"fs-id1165131906706\">a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131906711\">\n \t<dt>vector addition<\/dt>\n \t<dd id=\"fs-id1165135700056\">the sum of two vectors, found by adding corresponding components<\/dd>\n<\/dl>\n<\/div>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section you will:<\/p>\n<ul>\n<li>View vectors geometrically.<\/li>\n<li>Find magnitude and direction.<\/li>\n<li>Perform vector addition and scalar multiplication.<\/li>\n<li>Find the component form of a vector.<\/li>\n<li>Find the unit vector in the direction of\u2009[latex]v[\/latex].<\/li>\n<li>Perform operations with vectors in terms of\u2009[latex]i[\/latex]\u2009and\u2009[latex]j[\/latex].<\/li>\n<li>Find the dot product of two vectors.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135472124\">An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140\u00b0. A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_08_001\">(Figure)<\/a>. What are the ground speed and actual bearing of the plane?<\/p>\n<div id=\"Figure_08_08_001\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152829\/CNX_Precalc_Figure_08_08_001.jpg\" alt=\"Image of a plan flying SE at 140 degrees and the north wind blowing\" width=\"487\" height=\"462\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p>Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane\u2019s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let\u2019s examine the basics of vectors.<\/p>\n<div id=\"fs-id1165137737683\" class=\"bc-section section\">\n<h3>A Geometric View of Vectors<\/h3>\n<p id=\"fs-id1165135203508\">A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:<\/p>\n<ul>\n<li>Lower case, boldfaced type, with or without an arrow on top such as <strong>[latex]v,\\,\\,u,\\,\\,w,\\,\\,\\stackrel{\\to }{v},\\,\\,\\stackrel{\\to }{u},\\,\\stackrel{\\to }{w}.[\/latex]<\/strong><\/li>\n<li>Given initial point[latex]\\,P\\,[\/latex]and terminal point[latex]\\,Q,\\,[\/latex]a vector can be represented as[latex]\\,\\stackrel{\\to }{PQ}\\,.\\,\\,[\/latex]The arrowhead on top is what indicates that it is not just a line, but a directed line segment.<\/li>\n<li>Given an initial point of[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminal point[latex]\\,\\left(a,b\\right),\\,[\/latex]a vector may be represented as[latex]\u2329a,b\u232a.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137762492\">This last symbol [latex]\u2329a,b\u232a[\/latex] has special significance. It is called the standard position. The <span class=\"no-emphasis\">position vector<\/span> has an initial point [latex]\\left(0,0\\right)\\,[\/latex]and a terminal point[latex]\u2329a,b\u232a.[\/latex]To change any vector into the position vector, we think about the change in the <em>x<\/em>-coordinates and the change in the <em>y<\/em>-coordinates. Thus, if the initial point of a vector[latex]\\,\\stackrel{\\to }{CD}\\,[\/latex]is[latex]\\,C\\left({x}_{1},{y}_{1}\\right)\\,[\/latex]and the terminal point is[latex]\\,D\\left({x}_{2},{y}_{2}\\right),\\,[\/latex]then the position vector is found by calculating<\/p>\n<div id=\"fs-id1165135427119\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\stackrel{\\to }{AB}\\,=\\,\u2329{x}_{2}-{x}_{1},{y}_{2}-{y}_{1}\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\u2329a,b\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135160008\">In <a class=\"autogenerated-content\" href=\"#Figure_08_08_003\">(Figure)<\/a>, we see the original vector[latex]\\,\\stackrel{\\to }{CD}\\,[\/latex]and the position vector[latex]\\,\\stackrel{\\to }{AB}.[\/latex]<\/p>\n<div id=\"Figure_08_08_003\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152837\/CNX_Precalc_Figure_08_08_003.jpg\" alt=\"Plot of the original vector CD in blue and the position vector AB in orange extending from the origin.\" width=\"487\" height=\"290\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165135452050\">\n<h3>Properties of Vectors<\/h3>\n<p id=\"eip-id1165137910834\">A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at[latex]\\,\\left(0,0\\right)\\,[\/latex]and is identified by its terminal point[latex]\u2329a,b\u232a.[\/latex]<\/p>\n<\/div>\n<div id=\"Example_08_08_01\" class=\"textbox examples\">\n<div id=\"fs-id1165134568991\">\n<div id=\"fs-id1165134568994\">\n<h3>Find the Position Vector<\/h3>\n<p id=\"fs-id1165137736871\">Consider the vector whose initial point is[latex]\\,P\\left(2,3\\right)\\,[\/latex]and terminal point is[latex]\\,Q\\left(6,4\\right).\\,[\/latex]Find the position vector.<\/p>\n<\/div>\n<div id=\"fs-id1165137655654\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>The position vector is found by subtracting one <em>x<\/em>-coordinate from the other <em>x<\/em>-coordinate, and one <em>y<\/em>-coordinate from the other <em>y<\/em>-coordinate. Thus<\/p>\n<div id=\"fs-id1165137893431\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u23296-2,4-3\u232a\\hfill \\\\ \\,\\,\\,=\u23294,1\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137817747\">The position vector begins at[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminates at[latex]\\,\\left(4,1\\right).\\,[\/latex]The graphs of both vectors are shown in <a class=\"autogenerated-content\" href=\"#Figure_08_08_022\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_08_022\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152840\/CNX_Precalc_Figure_08_08_022.jpg\" alt=\"Plot of the original vector in blue and the position vector in orange extending from the origin.\" width=\"487\" height=\"349\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137532813\">We see that the position vector is[latex]\u23294,1\u232a.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137766874\">\n<div>\n<h3>Drawing a Vector with the Given Criteria and Its Equivalent Position Vector<\/h3>\n<p id=\"fs-id1165135390738\">Find the position vector given that vector<strong>[latex]\\,v\\,[\/latex]<\/strong>has an initial point at [latex]\\,\\left(-3,2\\right)\\,[\/latex]and a terminal point at[latex]\\,\\left(4,5\\right),\\,[\/latex]then graph both vectors in the same plane.<\/p>\n<\/div>\n<div id=\"fs-id1165137827504\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134041193\">The position vector is found using the following calculation:<\/p>\n<div id=\"fs-id1165134041196\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u23294-\\left(-3\\right),5-2\u232a\\hfill \\\\ \\text{ }=\u23297,3\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p>Thus, the position vector begins at[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminates at[latex]\\,\\left(7,3\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_08_004\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152847\/CNX_Precalc_Figure_08_08_004n.jpg\" alt=\"Plot of the two given vectors their same position vector.\" width=\"487\" height=\"328\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165135347493\"><\/details>\n<p><span id=\"fs-id1165135538542\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135169334\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_08_01\">\n<div id=\"fs-id1165137596741\">\n<p id=\"fs-id1165137596742\">Draw a vector<strong>[latex]\\,v\\,[\/latex]<\/strong>that connects from the origin to the point[latex]\\,\\left(3,5\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137834146\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137656713\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152849\/CNX_Precalc_Figure_08_08_006.jpg\" alt=\"A vector from the origin to (3,5) - a line with an arrow at the (3,5) endpoint.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137551413\" class=\"bc-section section\">\n<h3>Finding Magnitude and Direction<\/h3>\n<p id=\"fs-id1165134381525\">To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.<\/p>\n<div id=\"fs-id1165135209432\">\n<h3>Magnitude and Direction of a Vector<\/h3>\n<p id=\"fs-id1165137422586\">Given a position vector<strong>[latex]\\,v[\/latex]<\/strong>[latex]=\u2329a,b\u232a,[\/latex]the magnitude is found by[latex]|v|=\\sqrt{{a}^{2}+{b}^{2}}.[\/latex]The direction is equal to the angle formed with the <em>x<\/em>-axis, or with the <em>y<\/em>-axis, depending on the application. For a position vector, the direction is found by[latex]\\,\\mathrm{tan}\\,\\theta =\\left(\\frac{b}{a}\\right)\u21d2\\theta ={\\mathrm{tan}}^{-1}\\left(\\frac{b}{a}\\right),\\,[\/latex]as illustrated in <a class=\"autogenerated-content\" href=\"#Figure_08_08_017\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_08_017\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152902\/CNX_Precalc_Figure_08_08_017new.jpg\" alt=\"Standard plot of a position vector (a,b) with magnitude |v| extending into Q1 at theta degrees.\" width=\"487\" height=\"216\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137730380\">Two vectors <strong><em>v<\/em><\/strong> and <strong><em>u<\/em><\/strong> are considered equal if they have the same magnitude and the same direction. Additionally, if both vectors have the same position vector, they are equal.<\/p>\n<\/div>\n<div id=\"Example_08_08_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137619099\">\n<div>\n<h3>Finding the Magnitude and Direction of a Vector<\/h3>\n<p id=\"fs-id1165137838002\">Find the magnitude and direction of the vector with initial point[latex]\\,P\\left(-8,1\\right)\\,[\/latex]and terminal point[latex]\\,Q\\left(-2,-5\\right).[\/latex]Draw the vector.<\/p>\n<\/div>\n<div id=\"fs-id1165135413661\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135413663\">First, find the <span class=\"no-emphasis\">position vector<\/span>.<\/p>\n<div id=\"fs-id1165137433330\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}u=\u2329-2,-\\left(-8\\right),-5-1\u232a\\hfill \\\\ \\text{ }=\u23296,-6\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134466815\">We use the Pythagorean Theorem to find the magnitude.<\/p>\n<div id=\"fs-id1165134466819\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|u|=\\sqrt{{\\left(6\\right)}^{2}+{\\left(-6\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{72}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=6\\sqrt{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137428177\">The direction is given as<\/p>\n<div id=\"fs-id1165137428180\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{tan}\\,\\theta =\\frac{-6}{6}=-1\u21d2\\theta ={\\mathrm{tan}}^{-1}\\left(-1\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-45\u00b0\\hfill \\end{array}[\/latex]<\/div>\n<p>However, the angle terminates in the fourth quadrant, so we add 360\u00b0 to obtain a positive angle. Thus,[latex]\\,-45\u00b0+360\u00b0=315\u00b0.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_08_018\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152915\/CNX_Precalc_Figure_08_08_018.jpg\" alt=\"Plot of the position vector extending into Q4 from the origin with the magnitude 6rad2.\" width=\"487\" height=\"316\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165134254080\"><\/details>\n<p><span id=\"fs-id1165134196104\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135536154\">\n<div id=\"fs-id1165135536156\">\n<h3>Showing That Two Vectors Are Equal<\/h3>\n<p id=\"fs-id1165135349355\">Show that vector <strong><em>v<\/em><\/strong> with <span class=\"no-emphasis\">initial point<\/span> at[latex]\\,\\left(5,-3\\right)\\,[\/latex]and <span class=\"no-emphasis\">terminal point<\/span> at[latex]\\,\\left(-1,2\\right)\\,[\/latex]is equal to vector <strong><em>u<\/em><\/strong> with initial point at[latex]\\,\\left(-1,-3\\right)\\,[\/latex]and terminal point at[latex]\\,\\left(-7,2\\right).\\,[\/latex]Draw the position vector on the same grid as <strong><em>v<\/em><\/strong> and <strong><em>u<\/em><\/strong>. Next, find the magnitude and direction of each vector.<\/p>\n<\/div>\n<div id=\"fs-id1165137431395\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137431397\">As shown in <a class=\"autogenerated-content\" href=\"#Figure_08_08_005\">(Figure)<\/a>, draw the vector[latex]\\,v\\,[\/latex]starting at initial[latex]\\,\\left(5,-3\\right)\\,[\/latex]and terminal point[latex]\\,\\left(-1,2\\right).\\,[\/latex]Draw the vector[latex]\\,u\\,[\/latex]with initial point[latex]\\,\\left(-1,-3\\right)\\,[\/latex]and terminal point[latex]\\,\\left(-7,2\\right).\\,[\/latex]Find the standard position for each.<\/p>\n<p id=\"fs-id1165134314743\">Next, find and sketch the position vector for <strong><em>v<\/em><\/strong> and <strong><em>u<\/em><\/strong>. We have<\/p>\n<div id=\"fs-id1165137557695\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u2329-1-5,2-\\left(-3\\right)\u232a\\hfill \\\\ \\text{ }=\u2329-6,5\u232a\\hfill \\\\ \\hfill \\\\ u=\u2329-7-\\left(-1\\right),2-\\left(-3\\right)\u232a\\hfill \\\\ \\text{ }=\u2329-6,5\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135638538\">Since the position vectors are the same, <strong><em>v<\/em><\/strong> and <strong><em>u<\/em><\/strong> are the same.<\/p>\n<p id=\"fs-id1165135353002\">An alternative way to check for vector equality is to show that the magnitude and direction are the same for both vectors. To show that the magnitudes are equal, use the Pythagorean Theorem.<\/p>\n<div id=\"fs-id1165134224057\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|v|=\\sqrt{{\\left(-1-5\\right)}^{2}+{\\left(2-\\left(-3\\right)\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{{\\left(-6\\right)}^{2}+{\\left(5\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{36+25}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{61}\\hfill \\\\ |u|=\\sqrt{{\\left(-7-\\left(-1\\right)\\right)}^{2}+{\\left(2-\\left(-3\\right)\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{{\\left(-6\\right)}^{2}+{\\left(5\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{36+25}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{61}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135194193\">As the magnitudes are equal, we now need to verify the direction. Using the tangent function with the position vector gives<\/p>\n<div id=\"fs-id1165134339912\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{tan}\\,\\theta =-\\frac{5}{6}\u21d2\\theta ={\\mathrm{tan}}^{-1}\\left(-\\frac{5}{6}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-39.8\u00b0\\hfill \\end{array}[\/latex]<\/div>\n<p>However, we can see that the position vector terminates in the second quadrant, so we add[latex]\\,180\u00b0.\\,[\/latex]Thus, the direction is[latex]\\,-39.8\u00b0+180\u00b0=140.2\u00b0.[\/latex]<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152927\/CNX_Precalc_Figure_08_08_005n.jpg\" alt=\"Plot of the two given vectors their same position vector.\" width=\"487\" height=\"440\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165134569140\"><\/details>\n<p><span id=\"fs-id1165134108430\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137806748\" class=\"bc-section section\">\n<h3>Performing Vector Addition and Scalar Multiplication<\/h3>\n<p>Now that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector <strong>[latex]u[\/latex]<\/strong>[latex]=\u2329x,y\u232a[\/latex]as an arrow or directed line segment from the origin to the point[latex]\\,\\left(x,y\\right),\\,[\/latex]vectors can be situated anywhere in the plane. The sum of two vectors <strong><em>u<\/em><\/strong> and <strong><em>v<\/em><\/strong>, or vector addition, produces a third vector <strong><em>u<\/em><\/strong>+ <strong><em>v<\/em><\/strong>, the resultant vector.<\/p>\n<p id=\"fs-id1165135609392\">To find <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, we first draw the vector <strong><em>u<\/em><\/strong>, and from the terminal end of <strong><em>u<\/em><\/strong>, we drawn the vector <strong><em>v<\/em><\/strong>. In other words, we have the initial point of <strong><em>v<\/em><\/strong> meet the terminal end of <strong><em>u<\/em><\/strong>. This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong> is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of <strong><em>u<\/em><\/strong> to the end of <strong><em>v<\/em><\/strong> in a straight path, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_08_008\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_08_008\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152941\/CNX_Precalc_Figure_08_08_008.jpg\" alt=\"Diagrams of vector addition and subtraction.\" width=\"487\" height=\"149\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135415868\">Vector subtraction is similar to vector addition. To find <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>, view it as <strong><em>u<\/em><\/strong> + (\u2212<strong><em>v<\/em><\/strong>). Adding \u2212<strong><em>v<\/em><\/strong> is reversing direction of <strong><em>v<\/em><\/strong> and adding it to the end of <strong><em>u<\/em><\/strong>. The new vector begins at the start of <strong><em>u<\/em><\/strong> and stops at the end point of \u2212<strong><em>v<\/em><\/strong>. See <a class=\"autogenerated-content\" href=\"#Figure_08_08_009\">(Figure)<\/a> for a visual that compares vector addition and vector subtraction using <span class=\"no-emphasis\">parallelograms<\/span>.<\/p>\n<div id=\"Figure_08_08_009\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152958\/CNX_Precalc_Figure_08_08_009.jpg\" alt=\"Showing vector addition and subtraction with parallelograms. For addition, the base is u, the side is v, the diagonal connecting the start of the base to the end of the side is u+v. For subtraction, thetop is u, the side is -v, and the diagonal connecting the start of the top to the end of the side is u-v.\" width=\"487\" height=\"128\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"Example_08_08_05\" class=\"textbox examples\">\n<div id=\"fs-id1165135410202\">\n<div id=\"fs-id1165135410204\">\n<h3>Adding and Subtracting Vectors<\/h3>\n<p id=\"fs-id1165135410210\">Given <strong>[latex]u[\/latex]<\/strong>[latex]=\u23293,-2\u232a[\/latex]and<strong>[latex]v[\/latex]<\/strong>[latex]=\u2329-1,4\u232a,[\/latex]find two new vectors <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, and <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>.<\/p>\n<\/div>\n<div id=\"fs-id1165135407415\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135407417\">To find the sum of two vectors, we add the components. Thus,<\/p>\n<div id=\"fs-id1165135407421\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}u+v=\u23293,-2\u232a+\u2329-1,4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\u23293+\\left(-1\\right),-2+4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\u23292,2\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134081345\">See <a class=\"autogenerated-content\" href=\"#Figure_08_08_019\">(Figure)<\/a><strong>(a)<\/strong>.<\/p>\n<p id=\"fs-id1165134081354\">To find the difference of two vectors, add the negative components of<strong>[latex]\\,v\\,[\/latex]<\/strong>to<strong>[latex]\\,u.\\,[\/latex]<\/strong>Thus,<\/p>\n<div id=\"fs-id1165137832939\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}u+\\left(-v\\right)=\u23293,-2\u232a+\u23291,-4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\u23293+1,-2+\\left(-4\\right)\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\u23294,-6\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p>See <a class=\"autogenerated-content\" href=\"#Figure_08_08_019\">(Figure)<\/a><strong>(b).<\/strong><\/p>\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153012\/CNX_Precalc_Figure_08_08_019.jpg\" alt=\"Further diagrams of vector addition and subtraction.\" width=\"731\" height=\"292\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10. <\/strong>(a) Sum of two vectors (b) Difference of two vectors<\/figcaption><\/figure>\n<p id=\"fs-id1165134179704\"><\/details>\n<\/p>\n<div id=\"Figure_08_08_019\" class=\"medium\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137589359\" class=\"bc-section section\">\n<h3>Multiplying By a Scalar<\/h3>\n<p id=\"fs-id1165135472137\">While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.<\/p>\n<div id=\"fs-id1165134467706\">\n<h3>Scalar Multiplication<\/h3>\n<p id=\"fs-id1165135316119\">Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply <strong>[latex]v[\/latex]<\/strong>[latex]=\u2329a,b\u232a[\/latex] by [latex]k[\/latex], we have<\/p>\n<div id=\"fs-id1165133278770\" class=\"unnumbered aligncenter\">[latex]kv=\u2329ka,kb\u232a[\/latex]<\/div>\n<p id=\"fs-id1165137681116\">Only the magnitude changes, unless[latex]\\,k\\,[\/latex]is negative, and then the vector reverses direction.<\/p>\n<\/div>\n<div id=\"Example_08_08_06\" class=\"textbox examples\">\n<div id=\"fs-id1165134294910\">\n<div id=\"fs-id1165134294912\">\n<h3>Performing Scalar Multiplication<\/h3>\n<p id=\"fs-id1165135315522\">Given vector<strong>[latex]\\,v[\/latex]<\/strong>[latex]=\u23293,1\u232a,\\,[\/latex]find 3<strong><em>v<\/em><\/strong>, [latex]\\frac{1}{2}[\/latex]<strong>[latex]v,\\,[\/latex]<\/strong>and \u2212<strong><em>v<\/em><\/strong>.<\/p>\n<\/div>\n<div id=\"fs-id1165135580319\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135580321\">See <a class=\"autogenerated-content\" href=\"#Figure_08_08_007\">(Figure)<\/a> for a geometric interpretation. If<strong>[latex]\\,v[\/latex]<\/strong>[latex]=\u23293,1\u232a,[\/latex]then<\/p>\n<div id=\"eip-id3684825\" class=\"unnumbered\">[latex]\\begin{array}{l}\\,\\,3v=\u23293\\cdot 3,3\\cdot 1\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,=\u23299,3\u232a\\hfill \\\\ \\,\\frac{1}{2}v=\u2329\\frac{1}{2}\\cdot 3,\\frac{1}{2}\\cdot 1\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,=\u2329\\frac{3}{2},\\frac{1}{2}\u232a\\hfill \\\\ -v=\u2329-3,-1\u232a\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_08_08_007\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153015\/CNX_Precalc_Figure_08_08_007.jpg\" alt=\"Showing the effect of scaling a vector: 3x, 1x, .5x, and -1x. The 3x is three times as long, the 1x stays the same, the .5x halves the length, and the -1x reverses the direction of the vector but keeps the length the same. The rest keep the same direction; only the magnitude changes.\" width=\"487\" height=\"367\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134361434\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134153304\">Notice that the vector 3<strong><em>v<\/em><\/strong> is three times the length of <strong><em>v<\/em><\/strong>, [latex]\\frac{1}{2}[\/latex]<strong>[latex]v\\,[\/latex]<\/strong>is half the length of <strong><em>v<\/em><\/strong>, and \u2013<strong><em>v<\/em><\/strong> is the same length of <strong><em>v<\/em><\/strong>, but in the opposite direction.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165134159656\">\n<div id=\"fs-id1165134159657\">\n<p id=\"fs-id1165134159658\">Find the <span class=\"no-emphasis\">scalar multiple<\/span> 3<strong>[latex]u[\/latex]<\/strong> given <strong>[latex]u[\/latex]<\/strong>[latex]=\u23295,4\u232a.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135367782\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135367783\">[latex]3u=\u232915,12\u232a[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_07\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165135650641\">\n<h3>Using Vector Addition and Scalar Multiplication to Find a New Vector<\/h3>\n<p id=\"fs-id1165135434035\">Given <strong>[latex]u[\/latex]<\/strong>[latex]=\u23293,-2\u232a[\/latex]and<strong>[latex]v[\/latex]<\/strong>[latex]=\u2329-1,4\u232a,[\/latex]find a new vector <strong><em>w<\/em><\/strong> = 3<strong><em>u<\/em><\/strong> + 2<strong><em>v<\/em><\/strong>.<\/p>\n<\/div>\n<div id=\"fs-id1165137451791\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>First, we must multiply each vector by the scalar.<\/p>\n<div id=\"fs-id1165137451796\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}3u=3\u23293,-2\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\u23299,-6\u232a\\hfill \\\\ 2v=2\u2329-1,4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\u2329-2,8\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135208894\">Then, add the two together.<\/p>\n<div id=\"fs-id1165135208898\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}w=3u+2v\\hfill \\\\ \\,\\,\\,\\,\\,=\u23299,-6\u232a+\u2329-2,8\u232a\\hfill \\\\ \\,\\,\\,\\,\\,=\u23299-2,-6+8\u232a\\hfill \\\\ \\,\\,\\,\\,\\,=\u23297,2\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134066537\">So, <strong>[latex]w[\/latex]<\/strong>[latex]=\u23297,2\u232a.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137922516\" class=\"bc-section section\">\n<h3>Finding Component Form<\/h3>\n<p id=\"fs-id1165137401307\">In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the[latex]\\,x\\,[\/latex]direction, and the vertical component is the[latex]\\,y\\,[\/latex]direction. For example, we can see in the graph in <a class=\"autogenerated-content\" href=\"#Figure_08_08_020\">(Figure)<\/a> that the position vector[latex]\u23292,3\u232a[\/latex]comes from adding the vectors <strong><em>v<\/em><\/strong><sub>1<\/sub> and <strong><em>v<\/em><\/strong><sub>2<\/sub>. We have <strong><em>v<\/em><\/strong><sub>1<\/sub> with initial point[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminal point[latex]\\,\\left(2,0\\right).\\,[\/latex]<\/p>\n<div id=\"fs-id1165133045345\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{v}_{1}=\u23292-0,0-0\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\u23292,0\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135440228\">We also have <strong><em>v<\/em><\/strong><sub>2<\/sub> with initial point[latex]\\,\\left(0,0\\right)\\,[\/latex]and terminal point[latex]\\,\\left(0,\\,3\\right).\\,[\/latex]<\/p>\n<div id=\"eip-id4003269\" class=\"unnumbered\">[latex]\\begin{array}{l}{v}_{2}=\u23290-0,3-0\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,=\u23290,3\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135457923\">Therefore, the position vector is<\/p>\n<div id=\"fs-id1165135457926\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u23292+0,3+0\u232a\\hfill \\\\ \\,\\,\\,=\u23292,3\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137400265\">Using the Pythagorean Theorem, the magnitude of <strong><em>v<\/em><\/strong><sub>1<\/sub> is 2, and the magnitude of <strong><em>v<\/em><\/strong><sub>2<\/sub> is 3. To find the magnitude of <strong><em>v<\/em><\/strong>, use the formula with the position vector.<\/p>\n<div id=\"fs-id1165137849225\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}|v|=\\sqrt{|{v}_{1}{|}^{2}+|{v}_{2}{|}^{2}}\\hfill \\\\ \\begin{array}{l}\\,\\,\\,\\,\\,=\\sqrt{{2}^{2}+{3}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,=\\sqrt{13}\\hfill \\end{array}\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137387521\">The magnitude of <strong><em>v<\/em><\/strong> is[latex]\\,\\sqrt{13}.\\,[\/latex]To find the direction, we use the tangent function[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{y}{x}.[\/latex]<\/p>\n<div id=\"fs-id1165134224497\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{tan}\\,\\theta =\\frac{{v}_{2}}{{v}_{1}}\\hfill \\\\ \\mathrm{tan}\\,\\theta =\\frac{3}{2}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\theta ={\\mathrm{tan}}^{-1}\\left(\\frac{3}{2}\\right)=56.3\u00b0\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_08_08_020\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153029\/CNX_Precalc_Figure_08_08_020.jpg\" alt=\"Diagram of a vector in root position with its horizontal and vertical components.\" width=\"487\" height=\"289\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165134154507\">Thus, the magnitude of<strong>[latex]\\,v\\,[\/latex]<\/strong>is[latex]\\,\\sqrt{13}\\,[\/latex]and the direction is[latex]\\,{56.3}^{\\circ }[\/latex]off the horizontal.<\/p>\n<div id=\"Example_08_08_08\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137761654\">\n<h3>Finding the Components of the Vector<\/h3>\n<p id=\"fs-id1165134295643\">Find the components of the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>with initial point[latex]\\,\\left(3,2\\right)\\,[\/latex]and terminal point[latex]\\,\\left(7,4\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135253181\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135253183\">First find the standard position.<\/p>\n<div id=\"fs-id1165135253186\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\u23297-3,4-2\u232a\\hfill \\\\ \\,\\,\\,=\u23294,2\u232a\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137827401\">See the illustration in <a class=\"autogenerated-content\" href=\"#Figure_08_08_021\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_08_021\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153036\/CNX_Precalc_Figure_08_08_021.jpg\" alt=\"Diagram of a vector in root position with its horizontal (4,0) and vertical (0,2) components.\" width=\"487\" height=\"254\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137437225\">The horizontal component is <strong>[latex]{v}_{1}[\/latex]<\/strong>[latex]=\u23294,0\u232a\\,[\/latex]and the vertical component is<strong>[latex]\\,{v}_{2}[\/latex]<\/strong>[latex]=\u23290,2\u232a.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133299033\" class=\"bc-section section\">\n<h3>Finding the Unit Vector in the Direction of <em>v<\/em><\/h3>\n<p id=\"fs-id1165135634115\">In addition to finding a vector\u2019s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then preserve the direction of the original vector while simplifying calculations.<\/p>\n<p id=\"fs-id1165135190463\">Unit vectors are defined in terms of components. The horizontal unit vector is written as <strong>[latex]i[\/latex]<\/strong>[latex]=\u23291,0\u232a[\/latex]and is directed along the positive horizontal axis. The vertical unit vector is written as<strong>[latex]j[\/latex]<\/strong>[latex]=\u23290,1\u232a[\/latex]and is directed along the positive vertical axis. See <a class=\"autogenerated-content\" href=\"#Figure_08_08_011\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_08_011\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153043\/CNX_Precalc_Figure_08_08_011n.jpg\" alt=\"Plot showing the unit vectors i=91,0) and j=(0,1)\" width=\"487\" height=\"253\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 14.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165137838847\">\n<h3>The Unit Vectors<\/h3>\n<p id=\"fs-id1165134393822\">If<strong>[latex]\\,v\\,[\/latex]<\/strong>is a nonzero vector, then<strong>[latex]\\,\\frac{v}{|v|}\\,[\/latex]<\/strong>is a unit vector in the direction of<strong>[latex]\\,v.\\,[\/latex]<\/strong>Any vector divided by its magnitude is a unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.<\/p>\n<\/div>\n<div id=\"Example_08_08_09\" class=\"textbox examples\">\n<div id=\"fs-id1165137423314\">\n<div id=\"fs-id1165137423316\">\n<h3>Finding the Unit Vector in the Direction of <em>v<\/em><\/h3>\n<p id=\"fs-id1165134058261\">Find a unit vector in the same direction as <strong>[latex]v[\/latex]<\/strong>[latex]=\u2329-5,12\u232a.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137900009\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137900011\">First, we will find the magnitude.<\/p>\n<div id=\"fs-id1165137900014\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|v|=\\sqrt{{\\left(-5\\right)}^{2}+{\\left(12\\right)}^{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{25+144}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=\\sqrt{169}\\hfill \\\\ \\,\\,\\,\\,\\,\\,=13\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135570479\">Then we divide each component by[latex]\\,|v|,\\,[\/latex]which gives a unit vector in the same direction as <strong><em>v<\/em><\/strong>:<\/p>\n<div id=\"fs-id1165135193310\" class=\"unnumbered aligncenter\">[latex]\\frac{v}{|v|}=-\\frac{5}{13}i+\\frac{12}{13}j[\/latex]<\/div>\n<p>or, in component form<\/p>\n<div id=\"fs-id1165137653189\" class=\"unnumbered aligncenter\">[latex]\\frac{v}{|v|}=\u2329-\\frac{5}{13},\\frac{12}{13}\u232a[\/latex]<\/div>\n<p id=\"fs-id1165135154225\">See <a class=\"autogenerated-content\" href=\"#Figure_08_08_012\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_08_012\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153046\/CNX_Precalc_Figure_08_08_012.jpg\" alt=\"Plot showing the unit vector (-5\/13, 12\/13) in the direction of (-5, 12)\" width=\"487\" height=\"628\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 15.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137921617\">Verify that the magnitude of the unit vector equals 1. The magnitude of[latex]\\,-\\frac{5}{13}i+\\frac{12}{13}j\\,[\/latex]is given as<\/p>\n<div id=\"fs-id1165132960731\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\sqrt{{\\left(-\\frac{5}{13}\\right)}^{2}+{\\left(\\frac{12}{13}\\right)}^{2}}=\\sqrt{\\frac{25}{169}+\\frac{144}{169}}\\hfill \\\\ \\text{ }=\\sqrt{\\frac{169}{169}}=1\\hfill \\end{array}[\/latex]<\/div>\n<p>The vector <strong><em>u<\/em><\/strong>[latex]=\\frac{5}{13}[\/latex]<strong><em>i<\/em><\/strong>[latex]+\\frac{12}{13}[\/latex]<strong><em>j<\/em><\/strong> is the unit vector in the same direction as <strong><em>v<\/em><\/strong>[latex]=\u2329-5,12\u232a.[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137692582\" class=\"bc-section section\">\n<h3>Performing Operations with Vectors in Terms of <em>i<\/em> and <em>j <\/em><\/h3>\n<p id=\"fs-id1165135333199\">So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of <strong><em>i<\/em><\/strong> and <strong><em>j<\/em><\/strong>.<\/p>\n<div id=\"fs-id1165133075623\">\n<h3>Vectors in the Rectangular Plane<\/h3>\n<p id=\"fs-id1165133075632\">Given a vector<strong>[latex]\\,v\\,[\/latex]<\/strong>with initial point[latex]\\,P=\\left({x}_{1},{y}_{1}\\right)\\,[\/latex]and terminal point [latex]Q=\\left({x}_{2},{y}_{2}\\right),[\/latex] <strong><em>v<\/em><\/strong> is written as<\/p>\n<div id=\"fs-id1165135434061\" class=\"unnumbered aligncenter\">[latex]v=\\left({x}_{2}-{x}_{1}\\right)i+\\left({y}_{2}-{y}_{1}\\right)j[\/latex]<\/div>\n<p id=\"fs-id1165135180021\">The position vector from[latex]\\,\\left(0,0\\right)\\,[\/latex]to[latex]\\,\\left(a,b\\right),\\,[\/latex]where[latex]\\,\\left({x}_{2}-{x}_{1}\\right)=a\\,[\/latex]and[latex]\\,\\left({y}_{2}-{y}_{1}\\right)=b,\\,[\/latex]is written as <strong><em>v<\/em><\/strong> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em>. This vector sum is called a linear combination of the vectors <strong><em>i<\/em><\/strong> and <strong><em>j<\/em><\/strong>.<\/p>\n<p id=\"fs-id1165134151128\">The magnitude of <strong><em>v<\/em><\/strong> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> is given as[latex]\\,|v|=\\sqrt{{a}^{2}+{b}^{2}}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_08_010\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_08_010\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153051\/CNX_Precalc_Figure_08_08_010new.jpg\" alt=\"Plot showing vectors in rectangular coordinates in terms of i and j. The position vector v (in orange) extends from the origin to some point (a,b) in Q1. The horizontal (ai) and vertical (bj) components are shown.\" width=\"487\" height=\"237\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 16.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137640998\">\n<div id=\"fs-id1165137398636\">\n<h3>Writing a Vector in Terms of <em>i<\/em> and <em>j<\/em><\/h3>\n<p id=\"fs-id1165135195612\">Given a vector<strong>[latex]\\,v\\,[\/latex]<\/strong>with initial point[latex]\\,P=\\left(2,-6\\right)\\,[\/latex]and terminal point[latex]\\,Q=\\left(-6,6\\right),\\,[\/latex]write the vector in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165137415659\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137415662\">Begin by writing the general form of the vector. Then replace the coordinates with the given values.<\/p>\n<div id=\"fs-id1165137415666\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\\left({x}_{2}-{x}_{1}\\right)i+\\left({y}_{2}-{y}_{1}\\right)j\\hfill \\\\ \\,\\,\\,=\\left(-6-2\\right)i+\\left(6-\\left(-6\\right)\\right)j\\hfill \\\\ \\,\\,\\,=-8i+12j\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_11\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165134130103\">\n<h3>Writing a Vector in Terms of <em>i<\/em> and <em>j<\/em> Using Initial and Terminal Points<\/h3>\n<p id=\"fs-id1165137653576\">Given initial point[latex]\\,{P}_{1}=\\left(-1,3\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(2,7\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165132036992\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132036994\">Begin by writing the general form of the vector. Then replace the coordinates with the given values.<\/p>\n<div id=\"fs-id1165132036998\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v=\\left({x}_{2}-{x}_{1}\\right)i+\\left({y}_{2}-{y}_{1}\\right)j\\hfill \\\\ v=\\left(2-\\left(-1\\right)\\right)i+\\left(7-3\\right)j\\hfill \\\\ \\,\\,=3i+4j\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134309524\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_08_02\">\n<div id=\"fs-id1165133015819\">\n<p id=\"fs-id1165133015820\">Write the vector<strong>[latex]\\,u\\,[\/latex]<\/strong>with initial point[latex]\\,P=\\left(-1,6\\right)\\,[\/latex]and terminal point[latex]\\,Q=\\left(7,-5\\right)\\,[\/latex]in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165137726280\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137726281\">[latex]u=8i-11j[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134148227\" class=\"bc-section section\">\n<h3>Performing Operations on Vectors in Terms of <em>i<\/em> and <em>j<\/em><\/h3>\n<p id=\"fs-id1165135610243\">When vectors are written in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j,\\,[\/latex]<\/strong>we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.<\/p>\n<div id=\"fs-id1165135452303\">\n<h3>Adding and Subtracting Vectors in Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165137725223\">Given <strong><em>v<\/em><\/strong> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> and <strong><em>u<\/em><\/strong> = <em>c<strong>i<\/strong><\/em> + <em>d<strong>j<\/strong><\/em>, then<\/p>\n<div id=\"fs-id1165135332191\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}v+u=\\left(a+c\\right)i+\\left(b+d\\right)j\\\\ v-u=\\left(a-c\\right)i+\\left(b-d\\right)j\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_08_08_12\" class=\"textbox examples\">\n<div id=\"fs-id1165135481247\">\n<div id=\"fs-id1165135481249\">\n<h3>Finding the Sum of the Vectors<\/h3>\n<p id=\"fs-id1165135204368\">Find the sum of[latex]\\,{v}_{1}=2i-3j\\,[\/latex]and[latex]\\,{v}_{2}=4i+5j.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134547417\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134547419\">According to the formula, we have<\/p>\n<div id=\"fs-id1165135516771\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{v}_{1}+{v}_{2}=\\left(2+4\\right)i+\\left(-3+5\\right)j\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=6i+2j\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h3>Calculating the Component Form of a Vector: Direction<\/h3>\n<p id=\"fs-id1165135616272\">We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using[latex]\\,i\\,\\,\\text{and}\\,\\,j.\\,[\/latex]For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.<\/p>\n<p id=\"fs-id1165135515850\">Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with<strong>[latex]\\,|v|\\,[\/latex]<\/strong>replacing<strong>[latex]\\,r.[\/latex]<\/strong><\/p>\n<div id=\"fs-id1165134486704\">\n<h3>Vector Components in Terms of Magnitude and Direction<\/h3>\n<p id=\"fs-id1165137848784\">Given a position vector[latex]\\,v=\u2329x,y\u232a\\,[\/latex]and a direction angle[latex]\\,\\theta ,[\/latex]<\/p>\n<div id=\"fs-id1165133354263\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{lll}\\mathrm{cos}\\,\\theta =\\frac{x}{|v|}\\hfill & \\text{and}\\begin{array}{cc}& \\end{array}\\hfill & \\mathrm{sin}\\,\\theta =\\frac{y}{|v|}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,x=|v|\\mathrm{cos}\\,\\theta \\begin{array}{cc}& \\end{array}\\hfill & \\hfill & \\,\\,\\,\\,\\,\\,\\,y=|v|\\mathrm{sin}\\,\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135554288\">Thus,[latex]\\,v=xi+yj=|v|\\mathrm{cos}\\,\\theta i+|v|\\mathrm{sin}\\,\\theta j,\\,[\/latex]and magnitude is expressed as[latex]\\,|v|=\\sqrt{{x}^{2}+{y}^{2}}.[\/latex]<\/p>\n<\/div>\n<div id=\"Example_08_08_13\" class=\"textbox examples\">\n<div id=\"fs-id1165134280403\">\n<div id=\"fs-id1165134280405\">\n<h3>Writing a Vector in Terms of Magnitude and Direction<\/h3>\n<p id=\"fs-id1165134280410\">Write a vector with length 7 at an angle of 135\u00b0 to the positive<br \/>\n<em>x<\/em>-axis in terms of magnitude and direction.<\/p>\n<\/div>\n<div id=\"fs-id1165135421536\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135421538\">Using the conversion formulas[latex]\\,x=|v|\\mathrm{cos}\\,\\theta i\\,[\/latex]and[latex]\\,y=|v|\\mathrm{sin}\\,\\theta j,\\,[\/latex]we find that<\/p>\n<div id=\"fs-id1165135512708\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x=7\\mathrm{cos}\\left(135\u00b0\\right)i\\hfill \\\\ \\,\\,\\,=-\\frac{7\\sqrt{2}}{2}\\hfill \\\\ y=7\\mathrm{sin}\\left(135\u00b0\\right)j\\hfill \\\\ \\,\\,\\,=\\frac{7\\sqrt{2}}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165133389128\">This vector can be written as[latex]\\,v=7\\mathrm{cos}\\left(135\u00b0\\right)i+7\\mathrm{sin}\\left(135\u00b0\\right)j\\,[\/latex]or simplified as<\/p>\n<div id=\"fs-id1165135515872\" class=\"unnumbered aligncenter\">[latex]v=-\\frac{7\\sqrt{2}}{2}i+\\frac{7\\sqrt{2}}{2}j[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133095102\" class=\"textbox tryit\">\n<div id=\"ti_08_08_03\">\n<div id=\"fs-id1165133306703\">\n<p id=\"fs-id1165133306704\">A vector travels from the origin to the point[latex]\\,\\left(3,5\\right).\\,[\/latex]Write the vector in terms of magnitude and direction.<\/p>\n<\/div>\n<div id=\"fs-id1165131968594\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134391587\">[latex]v=\\sqrt{34}\\mathrm{cos}\\left(59\u00b0\\right)i+\\sqrt{34}\\mathrm{sin}\\left(59\u00b0\\right)j[\/latex]<\/p>\n<p>Magnitude =[latex]\\,\\sqrt{34}[\/latex]<\/p>\n<p id=\"fs-id1165135436242\">[latex]\\theta ={\\mathrm{tan}}^{-1}\\left(\\frac{5}{3}\\right)=59.04\u00b0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135381330\" class=\"bc-section section\">\n<h3>Finding the Dot Product of Two Vectors<\/h3>\n<p id=\"fs-id1165135381335\">As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the <em>dot product<\/em> and the <em>cross product<\/em>. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.<\/p>\n<p id=\"fs-id1165134085621\">The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.<\/p>\n<div id=\"fs-id1165134085624\">\n<h3>Dot Product<\/h3>\n<p id=\"fs-id1165134129884\">The dot product of two vectors[latex]\\,v=\u2329a,b\u232a\\,[\/latex]and[latex]\\,u=\u2329c,d\u232a\\,[\/latex]is the sum of the product of the horizontal components and the product of the vertical components.<\/p>\n<div id=\"fs-id1165137656787\" class=\"unnumbered aligncenter\">[latex]v\\cdot u=ac+bd[\/latex]<\/div>\n<p>To find the angle between the two vectors, use the formula below.<\/p>\n<div id=\"fs-id1165134156064\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cos}\\,\\theta =\\frac{v}{|v|}\\cdot \\frac{u}{|u|}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_08_08_14\" class=\"textbox examples\">\n<div id=\"fs-id1165135440032\">\n<div id=\"fs-id1165134149775\">\n<h3>Finding the Dot Product of Two Vectors<\/h3>\n<p>Find the dot product of<strong>[latex]\\,v=\u23295,12\u232a\\,[\/latex]<\/strong>and<strong>[latex]\\,u=\u2329-3,4\u232a.[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165137863042\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137863044\">Using the formula, we have<\/p>\n<div id=\"fs-id1165137863047\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}v\\cdot u=\u23295,12\u232a\\cdot \u2329-3,4\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5\\cdot \\left(-3\\right)+12\\cdot 4\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-15+48\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=33\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_15\" class=\"textbox examples\">\n<div id=\"fs-id1165134192846\">\n<div id=\"fs-id1165134192848\">\n<h3>Finding the Dot Product of Two Vectors and the Angle between Them<\/h3>\n<p id=\"fs-id1165134192853\">Find the dot product of <strong><em>v<\/em><\/strong><sub>1<\/sub> = 5<strong><em>i<\/em><\/strong> + 2<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong><sub>2<\/sub> = 3<strong><em>i<\/em><\/strong> + 7<strong><em>j<\/em><\/strong>. Then, find the angle between the two vectors.<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134330343\">Finding the dot product, we multiply corresponding components.<\/p>\n<div id=\"fs-id1165134330347\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{v}_{1}\\cdot {v}_{2}=\u23295,2\u232a\\cdot \u23293,7\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5\\cdot 3+2\\cdot 7\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=15+14\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=29\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135417616\">To find the angle between them, we use the formula[latex]\\,\\mathrm{cos}\\,\\theta =\\frac{v}{|v|}\\cdot \\frac{u}{|u|}.[\/latex]<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{v}{|v|}\\cdot \\frac{u}{|u|}=\u2329\\frac{5}{\\sqrt{29}}+\\frac{2}{\\sqrt{29}}\u232a\\cdot \u2329\\frac{3}{\\sqrt{58}}+\\frac{7}{\\sqrt{58}}\u232a\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{5}{\\sqrt{29}}\\cdot \\frac{3}{\\sqrt{58}}+\\frac{2}{\\sqrt{29}}\\cdot \\frac{7}{\\sqrt{58}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{15}{\\sqrt{1682}}+\\frac{14}{\\sqrt{1682}}=\\frac{29}{\\sqrt{1682}}\\hfill \\\\ \\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0.707107\\hfill \\\\ {\\mathrm{cos}}^{-1}\\left(0.707107\\right)=45\u00b0\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p>See <a class=\"autogenerated-content\" href=\"#Figure_08_08_014\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153113\/CNX_Precalc_Figure_08_08_014_Errata.jpg\" alt=\"Plot showing the two position vectors (3,7) and (5,2) and the 45 degree angle between them.\" width=\"488\" height=\"403\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 17.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165135181801\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_16\" class=\"textbox examples\">\n<div id=\"fs-id1165134041276\">\n<div id=\"fs-id1165134041278\">\n<h3>Finding the Angle between Two Vectors<\/h3>\n<p id=\"fs-id1165134041283\">Find the angle between[latex]\\,u=\u2329-3,4\u232a\\,[\/latex]and[latex]\\,v=\u23295,12\u232a.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135502770\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135502772\">Using the formula, we have<\/p>\n<div id=\"fs-id1165135502775\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\theta ={\\mathrm{cos}}^{-1}\\left(\\frac{u}{|u|}\\cdot \\frac{v}{|v|}\\right)\\hfill \\\\ \\left(\\frac{u}{|u|}\\cdot \\frac{v}{|v|}\\right)=\\frac{-3i+4j}{5}\\cdot \\frac{5i+12j}{13}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\left(-\\frac{3}{5}\\cdot \\frac{5}{13}\\right)+\\left(\\frac{4}{5}\\cdot \\frac{12}{13}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-\\frac{15}{65}+\\frac{48}{65}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{33}{65}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\theta ={\\mathrm{cos}}^{-1}\\left(\\frac{33}{65}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,={59.5}^{\\circ }\\hfill \\end{array}[\/latex]<\/div>\n<p>See <a class=\"autogenerated-content\" href=\"#Figure_08_08_013\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153121\/CNX_Precalc_Figure_08_08_013_Errata.jpg\" alt=\"Plot showing the two position vectors (-3,4) and (5,12) and the 59.5 degree angle between them.\" width=\"488\" height=\"628\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 18.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165135442401\"><\/details>\n<p><span id=\"fs-id1165135182874\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_08_17\" class=\"textbox examples\">\n<div id=\"fs-id1165137399741\">\n<div id=\"fs-id1165135503647\">\n<h3>Finding Ground Speed and Bearing Using Vectors<\/h3>\n<p id=\"fs-id1165135503653\">We now have the tools to solve the problem we introduced in the opening of the section.<\/p>\n<p id=\"fs-id1165135503656\">An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140\u00b0. A north wind (from north to south) is blowing at 16.2 miles per hour. What are the ground speed and actual bearing of the plane? See <a class=\"autogenerated-content\" href=\"#Figure_08_08_015\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_08_015\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153130\/CNX_Precalc_Figure_08_08_015.jpg\" alt=\"Image of a plan flying SE at 140 degrees and the north wind blowing.\" width=\"487\" height=\"462\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 19.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134043866\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134043868\">The ground speed is represented by[latex]\\,x\\,[\/latex]in the diagram, and we need to find the angle[latex]\\,\\alpha \\,[\/latex]in order to calculate the adjusted bearing, which will be[latex]\\,\\,140\u00b0+\\alpha \\,.[\/latex]<\/p>\n<p>Notice in <a class=\"autogenerated-content\" href=\"#Figure_08_08_015\">(Figure)<\/a>, that angle[latex]\\,BCO\\,[\/latex]must be equal to angle[latex]\\,AOC\\,[\/latex]by the rule of alternating interior angles, so angle[latex]\\,BCO\\,[\/latex]is 140\u00b0. We can find[latex]\\,x\\,[\/latex]by the Law of Cosines:<\/p>\n<div id=\"fs-id1165135181130\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{x}^{2}={\\left(16.2\\right)}^{2}+{\\left(200\\right)}^{2}-2\\left(16.2\\right)\\left(200\\right)\\mathrm{cos}\\left(140\u00b0\\right)\\hfill \\\\ {x}^{2}=45,226.41\\hfill \\\\ \\,\\,\\,x=\\sqrt{45,226.41}\\hfill \\\\ \\,\\,\\,x=212.7\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135453031\">The ground speed is approximately 213 miles per hour. Now we can calculate the bearing using the Law of Sines.<\/p>\n<div id=\"fs-id1165135453035\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{\\mathrm{sin}\\,\\alpha }{16.2}=\\frac{\\mathrm{sin}\\left(140\u00b0\\right)}{212.7}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\mathrm{sin}\\,\\alpha =\\frac{16.2\\mathrm{sin}\\left(140\u00b0\\right)}{212.7}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0.04896\\hfill \\\\ {\\mathrm{sin}}^{-1}\\left(0.04896\\right)=2.8\u00b0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134158931\">Therefore, the plane has a SE bearing of 140\u00b0+2.8\u00b0=142.8\u00b0. The ground speed is 212.7 miles per hour.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135442438\" class=\"precalculus media\">\n<p id=\"fs-id1165135442444\">Access these online resources for additional instruction and practice with vectors.<\/p>\n<ul>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/introvectors\">Introduction to Vectors<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/vectoroperation\">Vector Operations<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/unitvector\">The Unit Vector<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137686628\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137686633\">\n<li>The position vector has its initial point at the origin. See <a class=\"autogenerated-content\" href=\"#Example_08_08_01\">(Figure)<\/a>.<\/li>\n<li>If the position vector is the same for two vectors, they are equal. See <a class=\"autogenerated-content\" href=\"#Example_08_08_02\">(Figure)<\/a>.<\/li>\n<li>Vectors are defined by their magnitude and direction. See <a class=\"autogenerated-content\" href=\"#Example_08_08_03\">(Figure)<\/a>.<\/li>\n<li>If two vectors have the same magnitude and direction, they are equal. See <a class=\"autogenerated-content\" href=\"#Example_08_08_04\">(Figure)<\/a>.<\/li>\n<li>Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. See <a class=\"autogenerated-content\" href=\"#Example_08_08_05\">(Figure)<\/a>.<\/li>\n<li>Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. See <a class=\"autogenerated-content\" href=\"#Example_08_08_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_08_07\">(Figure)<\/a>.<\/li>\n<li>Vectors are comprised of two components: the horizontal component along the positive <em>x<\/em>-axis, and the vertical component along the positive <em>y<\/em>-axis. See <a class=\"autogenerated-content\" href=\"#Example_08_08_08\">(Figure)<\/a>.<\/li>\n<li>The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.<\/li>\n<li>The magnitude of a vector in the rectangular coordinate system is[latex]\\,|v|=\\sqrt{{a}^{2}+{b}^{2}}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_08_09\">(Figure)<\/a><strong>.<\/strong><\/li>\n<li>In the rectangular coordinate system, unit vectors may be represented in terms of <strong>[latex]i[\/latex]<\/strong> and <strong>[latex]j[\/latex]<\/strong> where<strong>[latex]\\,i\\,[\/latex]<\/strong>represents the horizontal component and<strong>[latex]\\,j\\,[\/latex]<\/strong>represents the vertical component. Then, <strong><em>v<\/em><\/strong> = a<strong><em>i<\/em><\/strong> + b<strong><em>j<\/em><\/strong>\u2009 is a scalar multiple of<strong>[latex]\\,v\\,[\/latex]<\/strong>by real numbers[latex]\\,a\\,\\text{and}\\,b.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_08_10\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_08_11\">(Figure)<\/a>.<\/li>\n<li>Adding and subtracting vectors in terms of <em>i<\/em> and <em>j<\/em> consists of adding or subtracting corresponding coefficients of <em>i<\/em> and corresponding coefficients of <em>j<\/em>. See <a class=\"autogenerated-content\" href=\"#Example_08_08_12\">(Figure)<\/a>.<\/li>\n<li>A vector <em>v<\/em> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> is written in terms of magnitude and direction as[latex]\\,v=|v|\\mathrm{cos}\\,\\theta i+|v|\\mathrm{sin}\\,\\theta j.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_08_13\">(Figure)<\/a>.<\/li>\n<li>The dot product of two vectors is the product of the<strong>[latex]\\,i\\,[\/latex]<\/strong>terms plus the product of the<strong>[latex]\\,j\\,[\/latex]<\/strong>terms. See <a class=\"autogenerated-content\" href=\"#Example_08_08_14\">(Figure)<\/a>.<\/li>\n<li>We can use the dot product to find the angle between two vectors. <a class=\"autogenerated-content\" href=\"#Example_08_08_15\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_08_16\">(Figure)<\/a>.<\/li>\n<li>Dot products are useful for many types of physics applications. See <a class=\"autogenerated-content\" href=\"#Example_08_08_17\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135404190\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135361194\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135361200\">\n<div id=\"fs-id1165135361202\">\n<p id=\"fs-id1165135361203\">What are the characteristics of the letters that are commonly used to represent vectors?<\/p>\n<\/div>\n<div id=\"fs-id1165135361206\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135361208\">lowercase, bold letter, usually[latex]\\,u,v,w[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134328318\">\n<div>\n<p>How is a vector more specific than a line segment?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135551174\">\n<div id=\"fs-id1165135551176\">\n<p id=\"fs-id1165135551178\">What are<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j,[\/latex]<\/strong>and what do they represent?<\/p>\n<\/div>\n<div id=\"fs-id1165134279576\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134279578\">They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279585\">\n<div id=\"fs-id1165135442412\">\n<p id=\"fs-id1165135442414\">What is component form?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135442420\">\n<div id=\"fs-id1165135442422\">\n<p id=\"fs-id1165135442424\">When a unit vector is expressed as[latex]\u2329a,b\u232a,[\/latex]which letter is the coefficient of the<strong>[latex]\\,i\\,[\/latex]<\/strong>and which the<strong>[latex]\\,j?[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165135309849\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135309851\">The first number always represents the coefficient of the[latex]\\,i,\\,[\/latex]and the second represents the[latex]\\,j.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134177056\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id1165137806553\">\n<div id=\"fs-id1165137806556\">\n<p id=\"fs-id1165137806558\">Given a vector with initial point[latex]\\,\\left(5,2\\right)\\,[\/latex]and terminal point[latex]\\,\\left(-1,-3\\right),\\,[\/latex]find an equivalent vector whose initial point is[latex]\\,\\left(0,0\\right).\\,[\/latex]Write the vector in component form[latex]\u2329a,b\u232a.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134318806\">\n<div id=\"fs-id1165134318807\">\n<p id=\"fs-id1165134318808\">Given a vector with initial point[latex]\\,\\left(-4,2\\right)\\,[\/latex]and terminal point[latex]\\,\\left(3,-3\\right),\\,[\/latex]find an equivalent vector whose initial point is[latex]\\,\\left(0,0\\right).\\,[\/latex]Write the vector in component form[latex]\u2329a,b\u232a.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133050481\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133050483\">[latex]\u30087,-5\u3009[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134475649\">\n<div id=\"fs-id1165134475650\">\n<p id=\"fs-id1165134475651\">Given a vector with initial point[latex]\\,\\left(7,-1\\right)\\,[\/latex]and terminal point[latex]\\,\\left(-1,-7\\right),\\,[\/latex]find an equivalent vector whose initial point is[latex]\\,\\left(0,0\\right).\\,[\/latex]Write the vector in component form[latex]\u2329a,b\u232a.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134149962\">For the following exercises, determine whether the two vectors<strong>[latex]\\,u\\,[\/latex]<\/strong>and<strong>[latex]\\,v\\,[\/latex]<\/strong>are equal, where<strong>[latex]\\,u\\,[\/latex]<\/strong>has an initial point[latex]\\,{P}_{1}\\,[\/latex]and a terminal point[latex]\\,{P}_{2}\\,[\/latex]and <strong>[latex]v[\/latex]<\/strong> has an initial point[latex]\\,{P}_{3}\\,[\/latex]and a terminal point[latex]\\,{P}_{4}[\/latex].<\/p>\n<div id=\"fs-id1165133354243\">\n<div id=\"fs-id1165133354244\">\n<p id=\"fs-id1165133354245\">[latex]{P}_{1}=\\left(5,1\\right),{P}_{2}=\\left(3,-2\\right),{P}_{3}=\\left(-1,3\\right),\\,[\/latex]and[latex]\\,{P}_{4}=\\left(9,-4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133022986\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133022988\">not equal<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133022992\">\n<div id=\"fs-id1165133022994\">\n<p id=\"fs-id1165135534902\">[latex]{P}_{1}=\\left(2,-3\\right),{P}_{2}=\\left(5,1\\right),{P}_{3}=\\left(6,-1\\right),\\,[\/latex]and[latex]\\,{P}_{4}=\\left(9,3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134032413\">\n<div id=\"fs-id1165134032414\">\n<p id=\"fs-id1165134573200\">[latex]{P}_{1}=\\left(-1,-1\\right),{P}_{2}=\\left(-4,5\\right),{P}_{3}=\\left(-10,6\\right),\\,[\/latex]and[latex]\\,{P}_{4}=\\left(-13,12\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137832972\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137832974\">equal<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832978\">\n<div id=\"fs-id1165137832980\">\n<p id=\"fs-id1165137832981\">[latex]{P}_{1}=\\left(3,7\\right),{P}_{2}=\\left(2,1\\right),{P}_{3}=\\left(1,2\\right),\\,[\/latex]and[latex]\\,{P}_{4}=\\left(-1,-4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133355961\">\n<div id=\"fs-id1165133355962\">\n<p id=\"fs-id1165133355963\">[latex]{P}_{1}=\\left(8,3\\right),{P}_{2}=\\left(6,5\\right),{P}_{3}=\\left(11,8\\right),\\,[\/latex]and[latex]{P}_{4}=\\left(9,10\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133341007\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133341010\">equal<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133341014\">\n<div id=\"fs-id1165133341015\">\n<p id=\"fs-id1165133341016\">Given initial point[latex]\\,{P}_{1}=\\left(-3,1\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(5,2\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135464840\">\n<div id=\"fs-id1165135464841\">\n<p id=\"fs-id1165135464842\">Given initial point[latex]\\,{P}_{1}=\\left(6,0\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-1,-3\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165135662547\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135662549\">[latex]7i-3j[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135499649\">For the following exercises, use the vectors <strong><em>u<\/em><\/strong> = <strong><em>i<\/em><\/strong> + 5<strong><em>j<\/em><\/strong>, <strong><em>v<\/em><\/strong> = \u22122<strong><em>i<\/em><\/strong>\u2212 3<strong><em>j<\/em><\/strong>,\u2009 and <strong><em>w<\/em><\/strong> = 4<strong><em>i<\/em><\/strong> \u2212 <strong><em>j<\/em><\/strong>.<\/p>\n<div id=\"fs-id1165135152214\">\n<div id=\"fs-id1165135152215\">\n<p id=\"fs-id1165135152216\">Find <strong><em>u<\/em><\/strong> + (<strong><em>v<\/em><\/strong> \u2212 <strong><em>w<\/em><\/strong>)<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135400265\">\n<div id=\"fs-id1165135400266\">\n<p id=\"fs-id1165135400267\">Find 4<strong><em>v<\/em><\/strong> + 2<strong><em>u<\/em><\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165133233962\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135190041\">[latex]-6i-2j[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134247986\">For the following exercises, use the given vectors to compute <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>, and 2<strong><em>u<\/em><\/strong> \u2212 3<strong><em>v<\/em><\/strong>.<\/p>\n<div id=\"fs-id1165134437218\">\n<div id=\"fs-id1165134437219\">\n<p id=\"fs-id1165134437220\">[latex]u=\u23292,-3\u232a,v=\u23291,5\u232a[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137900278\">\n<div id=\"fs-id1165137900279\">\n<p id=\"fs-id1165137900280\">[latex]u=\u2329-3,4\u232a,v=\u2329-2,1\u232a[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137642613\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137642615\">[latex]u+v=\u3008-5,5\u3009,u-v=\u3008-1,3\u3009,2u-3v=\u30080,5\u3009[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133300700\">\n<div id=\"fs-id1165133300701\">\n<p id=\"fs-id1165133300702\">Let <strong><em>v<\/em><\/strong> = \u22124<strong><em>i<\/em><\/strong> + 3<strong><em>j<\/em><\/strong>. Find a vector that is half the length and points in the same direction as<strong>[latex]\\,v.[\/latex]<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134401632\">\n<div id=\"fs-id1165134401634\">\n<p id=\"fs-id1165134401635\">Let <strong><em>v<\/em><\/strong> = 5<strong><em>i<\/em><\/strong> + 2<strong><em>j<\/em><\/strong>. Find a vector that is twice the length and points in the opposite direction as<strong>[latex]\\,v.[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165133366189\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133366191\">[latex]-10i\u20134j[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134173717\">For the following exercises, find a unit vector in the same direction as the given vector.<\/p>\n<div id=\"fs-id1165137643853\">\n<div id=\"fs-id1165137643854\">\n<p id=\"fs-id1165137643855\"><strong><em>a<\/em><\/strong> = 3<strong><em>i<\/em><\/strong> + 4<strong><em>j<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137883771\">\n<div>\n<p><strong><em>b<\/em><\/strong> = \u22122<strong><em>i<\/em><\/strong> + 5<strong><em>j<\/em><\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165137892399\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137892401\">[latex]-\\frac{2\\sqrt{29}}{29}i+\\frac{5\\sqrt{29}}{29}j[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134129835\">\n<p id=\"fs-id1165134129836\"><strong><em>c<\/em><\/strong> = 10<strong><em>i<\/em><\/strong> \u2013 <strong><em>j<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134381775\">\n<div id=\"fs-id1165134381776\">\n<p id=\"fs-id1165134381777\">[latex]d=-\\frac{1}{3}i+\\frac{5}{2}j[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133408795\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]-\\frac{2\\sqrt{229}}{229}i+\\frac{15\\sqrt{229}}{229}j[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134057517\">\n<div id=\"fs-id1165134057518\">\n<p id=\"fs-id1165134057519\"><strong><em>u<\/em><\/strong> = 100<strong><em>i<\/em><\/strong> + 200<strong><em>j<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137846165\">\n<div id=\"fs-id1165137846166\">\n<p id=\"fs-id1165137846167\"><strong><em>u<\/em><\/strong> = \u221214<strong><em>i<\/em><\/strong> + 2<strong><em>j<\/em><\/strong><\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]-\\frac{7\\sqrt{2}}{10}i+\\frac{\\sqrt{2}}{10}j[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135445726\">For the following exercises, find the magnitude and direction of the vector,[latex]\\,0\\le \\theta <2\\pi .[\/latex]<\/p>\n<div id=\"fs-id1165132035955\">\n<div id=\"fs-id1165132035956\">\n<p id=\"fs-id1165132035957\">[latex]\u23290,4\u232a[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134113808\">\n<div id=\"fs-id1165134113810\">\n<p id=\"fs-id1165134113811\">[latex]\u23296,5\u232a[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137894302\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137894304\">[latex]|v|=7.810,\\theta =39.806\u00b0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134544880\">\n<div id=\"fs-id1165137810093\">\n<p id=\"fs-id1165137810094\">[latex]\u23292,-5\u232a[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137835752\">\n<div id=\"fs-id1165137835753\">\n<p id=\"fs-id1165137835754\">[latex]\u2329-4,-6\u232a[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]|v|=7.211,\\theta =236.310\u00b0[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134351886\">\n<div id=\"fs-id1165134351887\">\n<p id=\"fs-id1165134351888\">Given <strong><em>u<\/em><\/strong> = 3<strong><em>i<\/em><\/strong> \u2212 4<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = \u22122<strong><em>i<\/em><\/strong> + 3<strong><em>j<\/em><\/strong>, calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134540065\">\n<div id=\"fs-id1165134540066\">\n<p id=\"fs-id1165134540067\">Given <strong><em>u<\/em><\/strong> = \u2212<strong><em>i<\/em><\/strong> \u2212 <strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = <strong><em>i<\/em><\/strong> + 5<strong><em>j<\/em><\/strong>, calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165134185467\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134185469\">[latex]-6[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134129815\">\n<div id=\"fs-id1165134129816\">\n<p id=\"fs-id1165134129817\">Given[latex]\\,u=\u2329-2,4\u232a\\,[\/latex]and[latex]\\,v=\u2329-3,1\u232a,\\,[\/latex]calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134192874\">\n<div id=\"fs-id1165134192875\">\n<p>Given <strong><em>u<\/em><\/strong>[latex]=\u2329-1,6\u232a[\/latex]and <strong><em>v<\/em><\/strong>[latex]=\u23296,-1\u232a,[\/latex]calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165131949016\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131949018\">[latex]-12[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133300653\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165134254430\">For the following exercises, given<strong>[latex]\\,v,\\,[\/latex]<\/strong>draw<strong>[latex]v,[\/latex]<\/strong>3<strong><em>v<\/em><\/strong> and[latex]\\,\\frac{1}{2}v.[\/latex]<\/p>\n<div id=\"fs-id1165135404720\">\n<div id=\"fs-id1165135404721\">\n<p id=\"fs-id1165135404722\">[latex]\u23292,-1\u232a[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135412891\">\n<div>[latex]\u2329-1,4\u232a[\/latex]<\/div>\n<div id=\"fs-id1165134554319\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165134554325\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153133\/CNX_Precalc_Figure_08_08_253.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135328732\">\n<div id=\"fs-id1165135328733\">\n<p id=\"fs-id1165135328734\">[latex]\u2329-3,-2\u232a[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135360308\">For the following exercises, use the vectors shown to sketch <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>, and 2<strong><em>u<\/em><\/strong>.<\/p>\n<div id=\"fs-id1165135347501\">\n<div id=\"fs-id1165135347502\"><span id=\"fs-id1165135347509\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153135\/CNX_Precalc_Figure_08_08_204.jpg\" alt=\"Plot of vectors u and v extending from the same origin point. In terms of that point, u goes to (1,1) and v goes to (-1,2).\" \/><\/span><\/div>\n<div id=\"fs-id1165134130859\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165134130866\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153141\/CNX_Precalc_Figure_08_08_205.jpg\" alt=\"Plot of u+v, u-v, and 2u based on the above vectors. In relation to the same origin point, u+v goes to (0,3), u-v goes to (2,-1), and 2u goes to (2,2).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133056304\">\n<div id=\"fs-id1165133056305\"><span id=\"fs-id1165133056311\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153143\/CNX_Precalc_Figure_08_08_206.jpg\" alt=\"Plot of vectors u and v extending from the same origin point. In terms of that point, u goes to (1,2) and v goes to (1,-1).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135203268\">\n<div id=\"fs-id1165135203269\"><span id=\"fs-id1165135203274\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153149\/CNX_Precalc_Figure_08_08_208.jpg\" alt=\"Plot of vectors u and v located head to tail. Take u's start point as the origin. In terms of that, u goes from the origin to (3,-2), and v goes from (3,-2) to (2,-3)\" \/><\/span><\/div>\n<div id=\"fs-id1165135203284\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135191577\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153151\/CNX_Precalc_Figure_08_08_209.jpg\" alt=\"Plot of vectors u+v, u-v, and 2u based on the above vectors.Given that u's start point was the origin, u+v starts at the origin and goes to (2,-3); u-v starts at the origin and goes to (4,-1); 2u goes from the origin to (6,-4).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135191588\">For the following exercises, use the vectors shown to sketch 2<strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>.<\/p>\n<div id=\"fs-id1165135347138\">\n<div id=\"fs-id1165135347139\"><span id=\"fs-id1165135347146\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153159\/CNX_Precalc_Figure_08_08_210.jpg\" alt=\"Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (3,1) and v goes from the origin to (2,-2).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134272782\">\n<div id=\"fs-id1165134272783\"><span id=\"fs-id1165134272790\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153202\/CNX_Precalc_Figure_08_08_212.jpg\" alt=\"Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (1,-2) and v goes from the origin to (-3,-2).\" \/><\/span><\/div>\n<div id=\"fs-id1165137696690\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153204\/CNX_Precalc_Figure_08_08_213.jpg\" alt=\"Plot of a single vector. Taking the start point of the vector as (0,0) from the above set up, the vector goes from the origin to (-1,-6).\" \/><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135509130\">For the following exercises, use the vectors shown to sketch <strong><em>u<\/em><\/strong> \u2212 3<strong><em>v<\/em><\/strong>.<\/p>\n<div id=\"fs-id1165134193570\">\n<div id=\"fs-id1165134193571\"><span id=\"fs-id1165133281363\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153207\/CNX_Precalc_Figure_08_08_214.jpg\" alt=\"Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (-4,0) and v goes from the origin to (1,-1).\" \/><\/span><\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134311986\"><span id=\"fs-id1165134311993\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153213\/CNX_Precalc_Figure_08_08_216.jpg\" alt=\"Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (1,2) and v goes from the origin to (-2,1).\" \/><\/span><\/div>\n<div id=\"fs-id1165135665485\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135665494\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153216\/CNX_Precalc_Figure_08_08_217.jpg\" alt=\"Vector extending from the origin to (7,5), taking the base as the origin.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165133266615\">For the following exercises, write the vector shown in component form.<\/p>\n<div id=\"fs-id1165133266619\">\n<div id=\"fs-id1165133266620\"><span id=\"fs-id1165134467613\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153226\/CNX_Precalc_Figure_08_08_218.jpg\" alt=\"Vector going from the origin to (-4,2).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134467625\">\n<div id=\"fs-id1165134467626\"><span id=\"fs-id1165133354213\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153235\/CNX_Precalc_Figure_08_08_219.jpg\" alt=\"Insert figure(table) alt text: Vector going from the origin to (4,1).\" \/><\/span><\/div>\n<div id=\"fs-id1165133354225\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134267740\">[latex]\u30084,1\u3009[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135649444\">\n<div id=\"fs-id1165135649445\">\n<p id=\"fs-id1165135649446\">Given initial point[latex]\\,{P}_{1}=\\left(2,1\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-1,2\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j,\\,[\/latex]<\/strong>then draw the vector on the graph.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133213855\">\n<div id=\"fs-id1165133213856\">\n<p id=\"fs-id1165133213857\">Given initial point[latex]\\,{P}_{1}=\\left(4,-1\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-3,2\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong>Draw the points and the vector on the graph.<\/p>\n<\/div>\n<div id=\"fs-id1165134179596\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134179598\">[latex]v=-7i+3j[\/latex]<\/p>\n<p><span id=\"fs-id1165137892136\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153237\/CNX_Precalc_Figure_08_08_221.jpg\" alt=\"Vector going from (4,-1) to (-3,2).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656840\">\n<div id=\"fs-id1165137656843\">\n<p id=\"fs-id1165137656845\">Given initial point[latex]\\,{P}_{1}=\\left(3,3\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-3,3\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong>Draw the points and the vector on the graph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135247487\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165135247492\">For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.<\/p>\n<div id=\"fs-id1165135247497\">\n<div id=\"fs-id1165135247498\">\n<p id=\"fs-id1165134043940\">[latex]|v|=6,\\theta =45\u00b0[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137942347\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137942349\">[latex]3\\sqrt{2}i+3\\sqrt{2}j[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135359683\">\n<div id=\"fs-id1165135359684\">\n<p id=\"fs-id1165135359685\">[latex]|v|=8,\\theta =220\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135181781\">\n<p id=\"fs-id1165135181782\">[latex]|v|=2,\\theta =300\u00b0[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137936571\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137936573\">[latex]i-\\sqrt{3}j[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134394479\">\n<div id=\"fs-id1165134394480\">\n<p id=\"fs-id1165134394482\">[latex]|v|=5,\\theta =135\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661647\">\n<div id=\"fs-id1165137661648\">\n<p id=\"fs-id1165137661649\">A 60-pound box is resting on a ramp that is inclined 12\u00b0. Rounding to the nearest tenth,<\/p>\n<ol id=\"fs-id1165135397073\" type=\"a\">\n<li>Find the magnitude of the normal (perpendicular) component of the force.<\/li>\n<li>Find the magnitude of the component of the force that is parallel to the ramp.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135397085\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135397087\">a. 58.7; b. 12.5<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135397090\">\n<div id=\"fs-id1165135397092\">\n<p id=\"fs-id1165135361159\">A 25-pound box is resting on a ramp that is inclined 8\u00b0. Rounding to the nearest tenth,<\/p>\n<ol id=\"fs-id1165135361164\" type=\"a\">\n<li>Find the magnitude of the normal (perpendicular) component of the force.<\/li>\n<li>Find the magnitude of the component of the force that is parallel to the ramp.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135361176\">\n<div id=\"fs-id1165134230559\">\n<p id=\"fs-id1165134230560\">Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of 27\u00b0 above the horizontal. Round to the nearest hundredth.<\/p>\n<\/div>\n<div id=\"fs-id1165134230567\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134230569\">[latex]x=7.13\\,[\/latex]pounds,[latex]\\,y=3.63\\,[\/latex]pounds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135320315\">\n<div id=\"fs-id1165135320316\">\n<p id=\"fs-id1165135320317\">Find the magnitude of the horizontal and vertical components of the vector with magnitude 4 pounds pointed in a direction of 127\u00b0 above the horizontal. Round to the nearest hundredth.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135320323\">\n<div id=\"fs-id1165135320324\">\n<p id=\"fs-id1165135320325\">Find the magnitude of the horizontal and vertical components of a vector with magnitude 5 pounds pointed in a direction of 55\u00b0 above the horizontal. Round to the nearest hundredth.<\/p>\n<\/div>\n<div id=\"fs-id1165134042375\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134042378\">[latex]x=2.87\\,[\/latex]pounds,[latex]\\,y=4.10\\,[\/latex]pounds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135452097\">\n<div id=\"fs-id1165135452098\">\n<p id=\"fs-id1165135452099\">Find the magnitude of the horizontal and vertical components of the vector with magnitude 1 pound pointed in a direction of 8\u00b0 above the horizontal. Round to the nearest hundredth.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135452106\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135452111\">\n<div id=\"fs-id1165134031246\">\n<p>A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?<\/p>\n<\/div>\n<div id=\"fs-id1165134031254\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134031256\">4.635 miles, 17.764\u00b0 N of E<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134031260\">\n<div id=\"fs-id1165134031261\">\n<p id=\"fs-id1165134031262\">A boat leaves the marina and sails 6 miles north, then 2 miles northeast. How far from the marina is the boat, and in what direction must it sail to head directly back to the marina?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135354908\">\n<div id=\"fs-id1165135354909\">\n<p id=\"fs-id1165135354910\">A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?<\/p>\n<\/div>\n<div id=\"fs-id1165135354915\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135354917\">17 miles. 10.318 miles<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135354921\">\n<div id=\"fs-id1165135354922\">\n<p id=\"fs-id1165134248817\">A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight home, how far would she have to walk?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137673347\">\n<div id=\"fs-id1165137673348\">\n<p id=\"fs-id1165137673349\">A man starts walking from home and walks 3 miles at 20\u00b0 north of west, then 5 miles at 10\u00b0 west of south, then 4 miles at 15\u00b0 north of east. If he walked straight home, how far would he have to the walk, and in what direction?<\/p>\n<\/div>\n<div id=\"fs-id1165137673356\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137673359\">Distance: 2.868. Direction: 86.474\u00b0 North of West, or 3.526\u00b0 West of North<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134116933\">\n<div id=\"fs-id1165134116934\">\n<p id=\"fs-id1165134116935\">A woman starts walking from home and walks 6 miles at 40\u00b0 north of east, then 2 miles at 15\u00b0 east of south, then 5 miles at 30\u00b0 south of west. If she walked straight home, how far would she have to walk, and in what direction?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134116942\">\n<div id=\"fs-id1165134116943\">\n<p id=\"fs-id1165134116944\">An airplane is heading north at an airspeed of 600 km\/hr, but there is a wind blowing from the southwest at 80 km\/hr. How many degrees off course will the plane end up flying, and what is the plane\u2019s speed relative to the ground?<\/p>\n<\/div>\n<div id=\"fs-id1165134370072\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134370074\">4.924\u00b0. 659 km\/hr<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134370078\">\n<div id=\"fs-id1165134370080\">\n<p>An airplane is heading north at an airspeed of 500 km\/hr, but there is a wind blowing from the northwest at 50 km\/hr. How many degrees off course will the plane end up flying, and what is the plane\u2019s speed relative to the ground?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134040498\">\n<div id=\"fs-id1165134040499\">\n<p id=\"fs-id1165134040500\">An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km\/hr. The plane flies with an airspeed of 550 km\/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?<\/p>\n<\/div>\n<div id=\"fs-id1165134040504\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134040507\">4.424\u00b0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134040511\">\n<div id=\"fs-id1165134040512\">\n<p id=\"fs-id1165134040513\">An airplane needs to head due north, but there is a wind blowing from the northwest at 80 km\/hr. The plane flies with an airspeed of 500 km\/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133306917\">\n<div id=\"fs-id1165133306918\">\n<p>As part of a video game, the point[latex]\\,\\left(5,7\\right)\\,[\/latex]is rotated counterclockwise about the origin through an angle of 35\u00b0. Find the new coordinates of this point.<\/p>\n<\/div>\n<div id=\"fs-id1165135582005\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134395208\">[latex]\\left(0.081,8.602\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134072229\">\n<div id=\"fs-id1165134072230\">\n<p id=\"fs-id1165134072231\">As part of a video game, the point[latex]\\,\\left(7,3\\right)\\,[\/latex]is rotated counterclockwise about the origin through an angle of 40\u00b0. Find the new coordinates of this point.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134398682\">\n<div id=\"fs-id1165134398683\">\n<p id=\"fs-id1165134398684\">Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn&#8217;t catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?<\/p>\n<\/div>\n<div id=\"fs-id1165134398690\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134398693\">21.801\u00b0, relative to the car\u2019s forward direction<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134398697\">\n<div id=\"fs-id1165134255038\">\n<p id=\"fs-id1165134183764\">Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 8 mph relative to the car, and the car is traveling 45 mph down the road. If one child doesn&#8217;t catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134183770\">\n<div id=\"fs-id1165134183771\">\n<p id=\"fs-id1165134183772\">A 50-pound object rests on a ramp that is inclined 19\u00b0. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.<\/p>\n<\/div>\n<div id=\"fs-id1165134183778\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134183781\">parallel: 16.28, perpendicular: 47.28 pounds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279250\">\n<div id=\"fs-id1165134279251\">\n<p id=\"fs-id1165134279252\">Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it upward, and 5 pounds acting on it 45\u00b0 from the horizontal. What single force is the resultant force acting on the body?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279259\">\n<div id=\"fs-id1165134279260\">\n<p id=\"fs-id1165134279261\">Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it \u2500135\u00b0 from the horizontal, and 5 pounds acting on it directed 150\u00b0 from the horizontal. What single force is the resultant force acting on the body?<\/p>\n<\/div>\n<div id=\"fs-id1165137920642\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137920644\">19.35 pounds, 231.54\u00b0 from the horizontal<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137920648\">\n<div id=\"fs-id1165137920650\">\n<p id=\"fs-id1165137920651\">The condition of equilibrium is when the sum of the forces acting on a body is the zero vector. Suppose a body has a force of 2 pounds acting on it to the right, 5 pounds acting on it upward, and 3 pounds acting on it 45\u00b0 from the horizontal. What single force is needed to produce a state of equilibrium on the body?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137920659\">\n<div id=\"fs-id1165133309247\">\n<p id=\"fs-id1165135364108\">Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30\u00b0 from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.<\/p>\n<\/div>\n<div id=\"fs-id1165133309254\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133309256\">5.1583 pounds, 75.8\u00b0 from the horizontal<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133309264\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1165134497721\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/aea1516b-dcb8-4853-a3b6-b15775067250\">Non-right Triangles: Law of Sines<\/a><\/h4>\n<p id=\"fs-id1165134497726\">For the following exercises, assume[latex]\\,\\alpha \\,[\/latex]is opposite side[latex]\\,a,\\beta \\,[\/latex]is opposite side[latex]\\,b,\\,[\/latex]and[latex]\\,\\gamma \\,[\/latex]is opposite side[latex]\\,c.\\,[\/latex]Solve each triangle, if possible. Round each answer to the nearest tenth.<\/p>\n<div id=\"fs-id1165135440338\">\n<div id=\"fs-id1165135440339\">\n<p id=\"fs-id1165135440340\">[latex]\\beta =50\u00b0,a=105,b=45[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134573204\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134573206\">Not possible<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134573212\">\n<p id=\"fs-id1165134573213\">[latex]\\alpha =43.1\u00b0,a=184.2,b=242.8[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134170057\">\n<div id=\"fs-id1165134170058\">\n<p id=\"fs-id1165134170059\">Solve the triangle.<\/p>\n<p><span id=\"fs-id1165134036684\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153244\/CNX_Precalc_Figure_08_08_223.jpg\" alt=\"Triangle with standard labels. Angle A is 36 degrees with opposite side a unknown. Angle B is 24 degrees with opposite side b = 16. Angle C and side c are unknown.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165134036696\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134036698\">[latex]C=120\u00b0,a=23.1,c=34.1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137936465\">\n<div id=\"fs-id1165137936466\">\n<p id=\"fs-id1165137936467\">Find the area of the triangle.<\/p>\n<p><span id=\"fs-id1165135317538\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153251\/CNX_Precalc_Figure_08_08_224.jpg\" alt=\"A triangle. One angle is 75 degrees with opposite side unknown. The adjacent sides to the 75 degree angle are 8 and 11.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135317550\">\n<div id=\"fs-id1165135384884\">\n<p id=\"fs-id1165135384885\">A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25\u00b0 and 49\u00b0, as shown in <a class=\"autogenerated-content\" href=\"#Image_08_08_225\">(Figure)<\/a>. Find the distance of the plane from point[latex]\\,A\\,[\/latex]and the elevation of the plane.<\/p>\n<div id=\"Image_08_08_225\" class=\"small\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153307\/CNX_Precalc_Figure_08_08_225.jpg\" alt=\"Diagram of a plane flying over a highway. It is to the left and above points A and B on the ground in that order. There is a horizontal line going through the plan parallel to the ground. The angle formed by the horizontal line, the plane, and the line from the plane to point B is 25 degrees. The angle formed by the horizontal line, the plane, and point A is 49 degrees.\" width=\"487\" height=\"201\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 20.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279590\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134279592\">distance of the plane from point[latex]\\,A:\\,[\/latex]2.2 km, elevation of the plane: 1.6 km<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134122825\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f59bb6c3-d618-4ba3-a4ab-97bd701e6957\">Non-right Triangles: Law of Cosines<\/a><\/h4>\n<div id=\"fs-id1165133078109\">\n<div id=\"fs-id1165133078110\">\n<p id=\"fs-id1165133078111\">Solve the triangle, rounding to the nearest tenth, assuming[latex]\\,\\alpha \\,[\/latex]is opposite side[latex]\\,a,\\beta \\,[\/latex]is opposite side[latex]\\,b,\\,[\/latex]and[latex]\\,\\gamma \\,[\/latex]s opposite side[latex]c:\\,a=4, b=6,c=8.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137676087\">\n<div id=\"fs-id1165137676088\">\n<p id=\"fs-id1165137676089\">Solve the triangle in <a class=\"autogenerated-content\" href=\"#Image_08_08_226\">(Figure)<\/a>, rounding to the nearest tenth.<\/p>\n<div id=\"Image_08_08_226\" class=\"small\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153310\/CNX_Precalc_Figure_08_08_226.jpg\" alt=\"A standardly labeled triangle. Angle A is 54 degrees with opposite side a unknown. Angle B is unknown with opposite side b=15. Angle C is unknown with opposite side C=13.\" width=\"487\" height=\"221\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 21.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134241045\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134178492\">[latex]B=71.0\u00b0,C=55.0\u00b0,a=12.8[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135407410\">\n<div id=\"fs-id1165135407411\">\n<p id=\"fs-id1165131962194\">Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131962197\">\n<div id=\"fs-id1165131962198\">\n<p id=\"fs-id1165131962199\">To find the distance between two cities, a satellite calculates the distances and angle shown in <a class=\"autogenerated-content\" href=\"#Image_08_08_227\">(Figure)<\/a> (not to scale). Find the distance between the cities. Round answers to the nearest tenth.<\/p>\n<div id=\"Image_08_08_227\" class=\"small\">\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153312\/CNX_Precalc_Figure_08_08_227.jpg\" alt=\"Diagram of a satellite above and to the right of two cities. The distance from the satellite to the closer city is 210 km. The distance from the satellite to the further city is 250 km. The angle formed by the closer city, the satellite, and the other city is 1.8 degrees.\" width=\"488\" height=\"264\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 22.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135329641\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135329643\">40.6 km<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134385712\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/c1e7a952-2a95-4202-b38b-126d1ea832fb\">Polar Coordinates<\/a><\/h4>\n<div id=\"fs-id1165134385717\">\n<div id=\"fs-id1165134385718\">\n<p id=\"fs-id1165134385719\">Plot the point with polar coordinates[latex]\\,\\left(3,\\frac{\\pi }{6}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541519\">\n<div id=\"fs-id1165135541520\">\n<p id=\"fs-id1165135541521\">Plot the point with polar coordinates[latex]\\,\\left(5,-\\frac{2\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134152554\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153315\/CNX_Precalc_Figure_08_08_229.jpg\" alt=\"Polar coordinate grid with a point plotted on the fifth concentric circle 2\/3 the way between pi and 3pi\/2 (closer to 3pi\/2).\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132914075\">\n<div id=\"fs-id1165132914076\">\n<p id=\"fs-id1165132914077\">Convert[latex]\\,\\left(6,-\\frac{3\\pi }{4}\\right)\\,[\/latex]to rectangular coordinates.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137701664\">\n<div id=\"fs-id1165137701665\">\n<p id=\"fs-id1165137701666\">Convert[latex]\\,\\left(-2,\\frac{3\\pi }{2}\\right)\\,[\/latex]to rectangular coordinates.<\/p>\n<\/div>\n<div id=\"fs-id1165135499576\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135499579\">[latex]\\,\\left(0,2\\right)\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131907335\">\n<div id=\"fs-id1165131907336\">\n<p id=\"fs-id1165131907337\">Convert[latex]\\left(7,-2\\right)[\/latex]to polar coordinates.<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135367681\">\n<p id=\"fs-id1165135367682\">Convert[latex]\\left(-9,-4\\right)[\/latex]<br \/>\nto polar coordinates.<\/p>\n<\/div>\n<div id=\"fs-id1165135440152\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135440154\">[latex]\\left(9.8489,203.96\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135640531\">For the following exercises, convert the given Cartesian equation to a polar equation.<\/p>\n<div id=\"fs-id1165135640534\">\n<div id=\"fs-id1165135640535\">\n<p id=\"fs-id1165135640536\">[latex]x=-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137760868\">\n<div id=\"fs-id1165137760869\">\n<p id=\"fs-id1165137760870\">[latex]{x}^{2}+{y}^{2}=64[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133001914\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133001916\">[latex]r=8[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134190679\">\n<div>\n<p id=\"fs-id1165134190681\">[latex]{x}^{2}+{y}^{2}=-2y[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135388841\">For the following exercises, convert the given polar equation to a Cartesian equation.<\/p>\n<div id=\"fs-id1165135388844\">\n<div id=\"fs-id1165135388845\">\n<p id=\"fs-id1165135388846\">[latex]r=7\\text{cos}\\,\\theta[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134138581\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134138583\">[latex]{x}^{2}+{y}^{2}=7x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135331692\">\n<div id=\"fs-id1165135331694\">\n<p id=\"fs-id1165135436286\">[latex]r=\\frac{-2}{4\\mathrm{cos}\\,\\theta +\\mathrm{sin}\\,\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165133078070\">For the following exercises, convert to rectangular form and graph.<\/p>\n<div id=\"fs-id1165133078073\">\n<div id=\"fs-id1165133078074\">\n<p id=\"fs-id1165133078075\">[latex]\\theta =\\frac{3\\pi }{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135521226\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135521228\">[latex]y=-x[\/latex]<\/p>\n<p><span id=\"fs-id1165137853290\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153324\/CNX_Precalc_Figure_08_08_230.jpg\" alt=\"Plot of the function y=-x in rectangular coordinates.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133437206\">\n<div id=\"fs-id1165133437207\">\n<p id=\"fs-id1165133437208\">[latex]r=5\\mathrm{sec}\\,\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/a63ed4d5-a31b-403b-9aaa-80329d8bcaa0\">Polar Coordinates: Graphs<\/a><\/h4>\n<p>For the following exercises, test each equation for symmetry.<\/p>\n<div>\n<div id=\"fs-id1165135618275\">[latex]r=4+4\\mathrm{sin}\\,\\theta[\/latex]<\/div>\n<div id=\"fs-id1165135519207\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135519209\">symmetric with respect to the line[latex]\\theta =\\frac{\\pi }{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135442491\">\n<div id=\"fs-id1165135442492\">\n<p id=\"fs-id1165135442493\">[latex]r=7[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134430440\">\n<p id=\"fs-id1165134430441\">Sketch a graph of the polar equation[latex]\\,r=1-5\\mathrm{sin}\\,\\theta .\\,[\/latex]Label the axis intercepts.<\/p>\n<\/div>\n<div id=\"fs-id1165134043892\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153327\/CNX_Precalc_Figure_08_08_232.jpg\" alt=\"Graph of the given polar equation - an inner loop lima\u00e7on.\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135403257\">\n<div id=\"fs-id1165135403258\">\n<p id=\"fs-id1165135403259\">Sketch a graph of the polar equation[latex]\\,r=5\\mathrm{sin}\\left(7\\theta \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137756884\">\n<div id=\"fs-id1165137756885\">\n<p id=\"fs-id1165137756886\">Sketch a graph of the polar equation[latex]\\,r=3-3\\mathrm{cos}\\,\\theta[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134534234\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153334\/CNX_Precalc_Figure_08_08_234.jpg\" alt=\"Graph of the given polar equation - a cardioid.\" \/><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135252219\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/5217ca36-7905-420a-b241-8d24df993e56\">Polar Form of Complex Numbers<\/a><\/h4>\n<p id=\"fs-id1165135252224\">For the following exercises, find the absolute value of each complex number.<\/p>\n<div id=\"fs-id1165135252227\">\n<div id=\"fs-id1165135252228\">[latex]-2+6i[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135209072\">\n<div id=\"fs-id1165135209073\">\n<p id=\"fs-id1165135209074\">[latex]4-\\text{\u200b}3i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135337142\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135337144\">5<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135337148\">Write the complex number in polar form.<\/p>\n<div id=\"fs-id1165135337152\">\n<div id=\"fs-id1165135337153\">\n<p id=\"fs-id1165135337154\">[latex]5+9i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135332336\">\n<div id=\"fs-id1165135332337\">\n<p id=\"fs-id1165135332338\">[latex]\\frac{1}{2}-\\frac{\\sqrt{3}}{2}\\text{\u200b}i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137707288\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137707290\">[latex]\\mathrm{cis}\\left(-\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165133023588\">For the following exercises, convert the complex number from polar to rectangular form.<\/p>\n<div id=\"fs-id1165133023591\">\n<div id=\"fs-id1165133023592\">\n<p id=\"fs-id1165134385585\">[latex]z=5\\mathrm{cis}\\left(\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134205830\">\n<div id=\"fs-id1165134205831\">\n<p id=\"fs-id1165134205832\">[latex]z=3\\mathrm{cis}\\left(40\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133318597\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133318599\">[latex]2.3+1.9i[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135360321\">For the following exercises, find the product[latex]\\,{z}_{1}{z}_{2}\\,[\/latex]in polar form.<\/p>\n<div id=\"fs-id1165135319994\">\n<div id=\"fs-id1165135319995\">\n<p>[latex]{z}_{1}=2\\mathrm{cis}\\left(89\u00b0\\right)[\/latex]<\/p>\n<p id=\"fs-id1165134159678\">[latex]{z}_{2}=5\\mathrm{cis}\\left(23\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134325188\">\n<div id=\"fs-id1165134325189\">\n<p id=\"fs-id1165134325191\">[latex]{z}_{1}=10\\mathrm{cis}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<p id=\"fs-id1165134313320\">[latex]{z}_{2}=6\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135702689\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135702692\">[latex]60\\mathrm{cis}\\left(\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165131883365\">For the following exercises, find the quotient[latex]\\,\\frac{{z}_{1}}{{z}_{2}}\\,[\/latex]in polar form.<\/p>\n<div id=\"fs-id1165135255380\">\n<div id=\"fs-id1165135255382\">\n<p>[latex]{z}_{1}=12\\mathrm{cis}\\left(55\u00b0\\right)[\/latex]<\/p>\n<p id=\"fs-id1165133221828\">[latex]{z}_{2}=3\\mathrm{cis}\\left(18\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133178324\">\n<div id=\"fs-id1165133178326\">\n<p id=\"fs-id1165133178328\">[latex]{z}_{1}=27\\mathrm{cis}\\left(\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n<p id=\"fs-id1165131968026\">[latex]{z}_{2}=9\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165131863139\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131863141\">[latex]3\\mathrm{cis}\\left(\\frac{4\\pi }{3}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134216196\">For the following exercises, find the powers of each complex number in polar form.<\/p>\n<div id=\"fs-id1165134216200\">\n<div id=\"fs-id1165134216201\">\n<p id=\"fs-id1165135332898\">Find[latex]\\,{z}^{4}\\,[\/latex]when[latex]\\,z=2\\mathrm{cis}\\left(70\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132059611\">\n<div id=\"fs-id1165132059612\">\n<p id=\"fs-id1165132059613\">Find[latex]\\,{z}^{2}\\,[\/latex]when[latex]\\,z=5\\mathrm{cis}\\left(\\frac{3\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133155816\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133155818\">[latex]25\\mathrm{cis}\\left(\\frac{3\\pi }{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135694330\">For the following exercises, evaluate each root.<\/p>\n<div id=\"fs-id1165135694334\">\n<div id=\"fs-id1165135694335\">\n<p id=\"fs-id1165135694336\">Evaluate the cube root of[latex]\\,z\\,[\/latex]when[latex]\\,z=64\\mathrm{cis}\\left(210\u00b0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133103838\">\n<div id=\"fs-id1165133103839\">\n<p id=\"fs-id1165133103840\">Evaluate the square root of[latex]\\,z\\,[\/latex]when[latex]\\,z=25\\mathrm{cis}\\left(\\frac{3\\pi }{2}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135701699\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135701701\">[latex]5\\mathrm{cis}\\left(\\frac{3\\pi }{4}\\right),5\\mathrm{cis}\\left(\\frac{7\\pi }{4}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137847075\">For the following exercises, plot the complex number in the complex plane.<\/p>\n<div id=\"fs-id1165137847078\">\n<div id=\"fs-id1165137847080\">\n<p id=\"fs-id1165137847081\">[latex]6-2i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132968119\">\n<div id=\"fs-id1165132968120\">\n<p id=\"fs-id1165132968121\">[latex]-1+3i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137937062\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137937070\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153337\/CNX_Precalc_Figure_08_08_236n.jpg\" alt=\"Plot of -1 + 3i in the complex plane (-1 along the real axis, 3 along the imaginary).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133237118\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/908896ca-db26-40f9-99de-79070485b4e7\">Parametric Equations<\/a><\/h4>\n<p id=\"fs-id1165133237123\">For the following exercises, eliminate the parameter[latex]\\,t\\,[\/latex]to rewrite the parametric equation as a Cartesian equation.<\/p>\n<div id=\"fs-id1165133349424\">\n<div>\n<p id=\"fs-id1165133349426\">[latex]\\{\\begin{array}{l}x(t)=3t-1\\hfill \\\\ y(t)=\\sqrt{t}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134234217\">\n<div id=\"fs-id1165134234218\">\n<p id=\"fs-id1165134234219\">[latex]\\{\\begin{array}{l}x(t)=-\\mathrm{cos}\\,t\\hfill \\\\ y(t)=2{\\mathrm{sin}}^{2}t \\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135407436\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135407438\">[latex]{x}^{2}+\\frac{1}{2}y=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135321170\">\n<div id=\"fs-id1165135321171\">\n<p id=\"fs-id1165135321172\">Parameterize (write a parametric equation for) each Cartesian equation by using[latex]\\,x\\left(t\\right)=a\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,y\\left(t\\right)=b\\mathrm{sin}\\,t\\,[\/latex]for[latex]\\,\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{16}=1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135512804\">\n<div id=\"fs-id1165135512805\">\n<p id=\"fs-id1165135512806\">Parameterize the line from[latex]\\,\\left(-2,3\\right)\\,[\/latex]to[latex]\\,\\left(4,7\\right)\\,[\/latex]so that the line is at[latex]\\,\\left(-2,3\\right)\\,[\/latex]at[latex]\\,t=0\\,[\/latex]and[latex]\\,\\left(4,7\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134177538\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134177540\">[latex]\\{\\begin{array}{l}x(t)=-2+6t\\hfill \\\\ y(t)=3+4t\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/38d919ed-1bee-4ae6-aed0-48285d870dab\">Parametric Equations: Graphs<\/a><\/h4>\n<p id=\"fs-id1165134177551\">For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.<\/p>\n<div id=\"fs-id1165134495161\">\n<div id=\"fs-id1165134495162\">\n<p id=\"fs-id1165134495163\">[latex]\\{\\begin{array}{l}x(t)=3{t}^{2}\\hfill \\\\ y(t)=2t-1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135455023\">\n<div id=\"fs-id1165135455024\">\n<p id=\"fs-id1165135455026\">[latex]\\{\\begin{array}{l}x(t)={e}^{t}\\hfill \\\\ y(t)=-2{e}^{5\\,t}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134272725\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135609213\">[latex]y=-2{x}^{5}[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153340\/CNX_Precalc_Figure_08_08_238.jpg\" alt=\"Plot of the given parametric equations.\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134185411\">\n<div id=\"fs-id1165134534274\">\n<p id=\"fs-id1165134534275\">[latex]\\{\\begin{array}{l}x(t)=3\\mathrm{cos}\\,t\\hfill \\\\ y(t)=2\\mathrm{sin}\\,t\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165131962229\">\n<p id=\"fs-id1165131962230\">A ball is launched with an initial velocity of 80 feet per second at an angle of 40\u00b0 to the horizontal. The ball is released at a height of 4 feet above the ground.<\/p>\n<ol id=\"fs-id1165131962236\" type=\"a\">\n<li>Find the parametric equations to model the path of the ball.<\/li>\n<li>Where is the ball after 3 seconds?<\/li>\n<li>How long is the ball in the air?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135472893\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165135472895\" type=\"a\">\n<li>[latex]\\{\\begin{array}{l}x(t)=(80\\mathrm{cos}(40\u00b0))t\\\\ y(t)=-16{t}^{2}+(80\\mathrm{sin}(40\u00b0))t+4\\end{array}[\/latex]<\/li>\n<li>The ball is 14 feet high and 184 feet from where it was launched.<\/li>\n<li>3.3 seconds<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134356905\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/907cf72a-9d36-4043-98b4-3d251188ac6f\">Vectors<\/a><\/h4>\n<p id=\"fs-id1165135673959\">For the following exercises, determine whether the two vectors,<strong>[latex]\\,u\\,[\/latex]<\/strong>and<strong>[latex]\\,v,\\,[\/latex]<\/strong>are equal, where<strong>[latex]\\,u\\,[\/latex]<\/strong>has an initial point[latex]\\,{P}_{1}\\,[\/latex]and a terminal point[latex]\\,{P}_{2},\\,[\/latex]and<strong>[latex]\\,v\\,[\/latex]<\/strong>has an initial point[latex]\\,{P}_{3}\\,[\/latex]and a terminal point[latex]\\,{P}_{4}.[\/latex]<\/p>\n<div>\n<div>\n<p id=\"fs-id1165134534170\">[latex]{P}_{1}=\\left(-1,4\\right),{P}_{2}=\\left(3,1\\right),{P}_{3}=\\left(5,5\\right)[\/latex]and[latex]\\,{P}_{4}=\\left(9,2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137580687\">\n<div id=\"fs-id1165137580688\">\n<p id=\"fs-id1165137580690\">[latex]{P}_{1}=\\left(6,11\\right),{P}_{2}=\\left(-2,8\\right),{P}_{3}=\\left(0,-1\\right)\\,[\/latex]and[latex]\\,{P}_{4}=\\left(-8,2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137780030\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137780032\">not equal<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135532362\">For the following exercises, use the vectors<strong>[latex]\\,u=2i-j\\text{,}v=4i-3j\\text{,}\\,[\/latex]<\/strong>and<strong>[latex]\\,w=-2i+5j\\,[\/latex]<\/strong>to evaluate the expression.<\/p>\n<div id=\"fs-id1165137734400\">\n<div id=\"fs-id1165137734401\">\n<p id=\"fs-id1165137734402\"><strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131886725\">\n<div id=\"fs-id1165131886726\">\n<p id=\"fs-id1165131886727\">2<strong><em>v<\/em><\/strong> \u2212 <strong><em>u<\/em><\/strong> + <strong><em>w<\/em><\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165133359377\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133359379\">4<strong><em>i<\/em><\/strong><\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137895109\">For the following exercises, find a unit vector in the same direction as the given vector.<\/p>\n<div id=\"fs-id1165137895112\">\n<div id=\"fs-id1165137895113\">\n<p id=\"fs-id1165137895114\"><strong><em>a<\/em><\/strong> = 8<strong><em>i<\/em><\/strong> \u2212 6<strong><em>j<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137938385\">\n<div id=\"fs-id1165137938386\">\n<p id=\"fs-id1165137938387\"><strong><em>b<\/em><\/strong> = \u22123<strong><em>i<\/em><\/strong> \u2212 <strong><em>j<\/em><\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165135351469\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135351471\">[latex]-\\frac{3\\sqrt{10}}{10}[\/latex]<strong><em>i<\/em><\/strong>[latex]-\\frac{\\sqrt{10}}{10}[\/latex]<strong><em>j<\/em><\/strong><\/p>\n<\/details>\n<\/div>\n<\/div>\n<p>For the following exercises, find the magnitude and direction of the vector.<\/p>\n<div id=\"fs-id1165135252126\">\n<div>\n<p id=\"fs-id1165135252128\">[latex]\u23296,-2\u232a[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135189796\">\n<div id=\"fs-id1165135189797\">\n<p id=\"fs-id1165135189798\">[latex]\u2329-3,-3\u232a[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165132936410\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132936412\">Magnitude:[latex]\\,3\\sqrt{2},\\,[\/latex]Direction:[latex]\\text{225\u00b0}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165132934428\">For the following exercises, calculate<strong>[latex]\\,u\\cdot v\\text{.}[\/latex]<\/strong><\/p>\n<div id=\"fs-id1165134179615\">\n<div id=\"fs-id1165134179616\">\n<p id=\"fs-id1165134179618\"><strong><em>u<\/em><\/strong> = \u22122<strong><em>i<\/em><\/strong> + <strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = 3<strong><em>i<\/em><\/strong> + 7<strong><em>j<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134248828\">\n<div id=\"fs-id1165134248829\">\n<p id=\"fs-id1165134248830\"><strong><em>u<\/em><\/strong> = <strong><em>i<\/em><\/strong> + 4<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = 4<strong><em>i<\/em><\/strong> + 3<strong><em>j<\/em><\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165137832740\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137832743\">[latex]\\text{16}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832747\">\n<div id=\"fs-id1165137832748\">\n<p id=\"fs-id1165137832749\">Given <strong><em>v<\/em><\/strong>[latex]=\u3008-3,4\u3009[\/latex]draw <strong><em>v<\/em><\/strong>, 2<strong><em>v<\/em><\/strong>, and [latex]\\,\\frac{1}{2}[\/latex]<strong><em>v<\/em><\/strong>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135694955\">\n<div id=\"fs-id1165135694956\">\n<p id=\"fs-id1165135694957\">Given the vectors shown in <a class=\"autogenerated-content\" href=\"#Image_08_08_241\">(Figure)<\/a>, sketch <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong> and 3<strong><em>v<\/em><\/strong>.<\/p>\n<div id=\"Image_08_08_241\" class=\"small\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153352\/CNX_Precalc_Figure_08_08_241.jpg\" alt=\"Diagram of vectors v, 2v, and 1\/2 v. The 2v vector is in the same direction as v but has twice the magnitude. The 1\/2 v vector is in the same direction as v but has half the magnitude.\" width=\"487\" height=\"323\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 23.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135532583\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153354\/CNX_Precalc_Figure_08_08_242.jpg\" alt=\"Diagram of vectors u and v. Taking u's starting point as the origin, u goes from the origin to (4,1), and v goes from (4,1) to (6,0).\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132957216\">\n<div id=\"fs-id1165132957217\">\n<p id=\"fs-id1165132957218\">Given initial point[latex]\\,{P}_{1}=\\left(3,2\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-5,-1\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,\\,i\\,\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong>Draw the points and the vector on the graph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135179890\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<div id=\"fs-id1165135179893\">\n<div id=\"fs-id1165135179894\">\n<p id=\"fs-id1165135179895\">Assume[latex]\\,\\alpha \\,[\/latex]is opposite side[latex]\\,a,\\beta \\,[\/latex]is opposite side[latex]\\,b,\\,[\/latex]and[latex]\\,\\gamma \\,[\/latex]is opposite side[latex]\\,c.\\,[\/latex]Solve the triangle, if possible, and round each answer to the nearest tenth, given[latex]\\,\\beta =68\u00b0,b=21,c=16.[\/latex]<\/p>\n<\/div>\n<div>[latex]\\alpha =67.1\u00b0,\\gamma =44.9\u00b0,a=20.9[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134230590\">\n<div id=\"fs-id1165134230591\">\n<p id=\"fs-id1165134230592\">Find the area of the triangle in <a class=\"autogenerated-content\" href=\"#Image_08_08_244\">(Figure)<\/a>. Round each answer to the nearest tenth.<\/p>\n<div id=\"Image_08_08_244\" class=\"small\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153357\/CNX_Precalc_Figure_08_08_244.jpg\" alt=\"A triangle. One angle is 60 degrees with opposite side 6.25. The other two sides are 5 and 7.\" width=\"487\" height=\"165\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 24.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134301539\">\n<div id=\"fs-id1165134301540\">\n<p id=\"fs-id1165134301541\">A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15\u00b0 to the right of his original course, and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?<\/p>\n<\/div>\n<div id=\"fs-id1165134301549\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134301551\">[latex]\\text{1712 miles}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134086133\">\n<p id=\"fs-id1165134086134\">Convert[latex]\\,\\left(2,2\\right)\\,[\/latex]<br \/>\nto polar coordinates, and then plot the point.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133200949\">\n<div id=\"fs-id1165133200950\">\n<p id=\"fs-id1165133200951\">Convert[latex]\\,\\left(2,\\frac{\\pi }{3}\\right)\\,[\/latex]to rectangular coordinates.<\/p>\n<\/div>\n<div id=\"fs-id1165135473730\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135473732\">[latex]\\left(1,\\sqrt{3}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134301575\">\n<div id=\"fs-id1165134301576\">\n<p id=\"fs-id1165134301578\">Convert the polar equation to a Cartesian equation:[latex]\\,{x}^{2}+{y}^{2}=5\\mathrm{y.}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135459827\">\n<p>Convert to rectangular form and graph:[latex]r=-3\\mathrm{csc}\\,\\theta .[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133359307\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133359309\">[latex]y=-3[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153359\/CNX_Precalc_Figure_08_08_246.jpg\" alt=\"Plot of the given equation in rectangular form - line y=-3.\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134085542\">\n<div id=\"fs-id1165134085543\">\n<p id=\"fs-id1165134085544\">Test the equation for symmetry:[latex]\\,r=-4\\mathrm{sin}\\left(2\\theta \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133363596\">\n<div id=\"fs-id1165133363598\">\n<p id=\"fs-id1165133363599\">Graph[latex]\\,r=3+3\\mathrm{cos}\\,\\theta .[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134081559\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153407\/CNX_Precalc_Figure_08_08_247.jpg\" alt=\"Graph of the given equations - a cardioid.\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134081170\">\n<div id=\"fs-id1165134081172\">\n<p id=\"fs-id1165134081174\">Graph[latex]\\,r=3-5\\text{sin}\\,\\theta .[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134532849\">\n<div id=\"fs-id1165134532851\">\n<p id=\"fs-id1165134532853\">Find the absolute value of the complex number<br \/>\n[latex]5-9i.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165131841660\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131841662\">[latex]\\sqrt{106}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134038130\">\n<div id=\"fs-id1165134038131\">\n<p id=\"fs-id1165134038132\">Write the complex number in polar form:[latex]\\,4+i\\text{.}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165134070883\">Convert the complex number from polar to rectangular form:[latex]\\,z=5\\text{cis}\\left(\\frac{2\\pi }{3}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137734329\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137734331\">[latex]\\frac{-5}{2}+i\\frac{5\\sqrt{3}}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135426348\">Given[latex]\\,{z}_{1}=8\\mathrm{cis}\\left(36\u00b0\\right)\\,[\/latex]and[latex]\\,{z}_{2}=2\\mathrm{cis}\\left(15\u00b0\\right),[\/latex]evaluate each expression.<\/p>\n<div id=\"fs-id1165134036642\">\n<div id=\"fs-id1165134036644\">\n<p id=\"fs-id1165134036646\">[latex]{z}_{1}{z}_{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135662488\">\n<div id=\"fs-id1165135662490\">\n<p id=\"fs-id1165135662492\">[latex]\\frac{{z}_{1}}{{z}_{2}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135653907\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135653909\">[latex]4\\mathrm{cis}\\left(21\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135305744\">\n<div id=\"fs-id1165135305746\">\n<p id=\"fs-id1165135305748\">[latex]{\\left({z}_{2}\\right)}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134430302\">\n<div id=\"fs-id1165134430304\">\n<p id=\"fs-id1165134430306\">[latex]\\sqrt{{z}_{1}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135537343\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135537345\">[latex]2\\sqrt{2}\\mathrm{cis}\\left(18\u00b0\\right),2\\sqrt{2}\\mathrm{cis}\\left(198\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132955611\">\n<div id=\"fs-id1165132955612\">\n<p id=\"fs-id1165132955613\">Plot the complex number[latex]\\,-5-i\\,[\/latex]in the complex plane.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135432745\">\n<div id=\"fs-id1165135432746\">\n<p id=\"fs-id1165135432747\">Eliminate the parameter[latex]\\,t\\,[\/latex]to rewrite the following parametric equations as a Cartesian equation: [latex]\\,\\{\\begin{array}{l}x(t)=t+1\\hfill \\\\ y(t)=2{t}^{2}\\hfill \\end{array}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134179578\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134179580\">[latex]y=2{\\left(x-1\\right)}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135342087\">\n<div id=\"fs-id1165135342089\">\n<p id=\"fs-id1165135342091\">Parameterize (write a parametric equation for) the following Cartesian equation by using[latex]\\,x\\left(t\\right)=a\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,y\\left(t\\right)=b\\mathrm{sin}\\,t:[\/latex][latex]\\frac{{x}^{2}}{36}+\\frac{{y}^{2}}{100}=1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135359801\">\n<div id=\"fs-id1165135359803\">\n<p id=\"fs-id1165135359805\">Graph the set of parametric equations and find the Cartesian equation:[latex]\\,\\{\\begin{array}{l}x(t)=-2\\mathrm{sin}\\,t\\hfill \\\\ y(t)=5\\mathrm{cos}\\,t\\hfill \\end{array}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135636921\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153409\/CNX_Precalc_Figure_08_08_250.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134053928\">\n<div id=\"fs-id1165137921477\">\n<p id=\"fs-id1165137921480\">A ball is launched with an initial velocity of 95 feet per second at an angle of 52\u00b0 to the horizontal. The ball is released at a height of 3.5 feet above the ground.<\/p>\n<ol id=\"fs-id1165137921485\" type=\"a\">\n<li>Find the parametric equations to model the path of the ball.<\/li>\n<li>Where is the ball after 2 seconds?<\/li>\n<li>How long is the ball in the air?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137921500\">For the following exercises, use the vectors <strong><em>u<\/em><\/strong> = <strong><em>i<\/em><\/strong> \u2212 3<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong> = 2<strong><em>i<\/em><\/strong> + 3<strong><em>j<\/em><\/strong>.<\/p>\n<div id=\"fs-id1165134257477\">\n<div id=\"fs-id1165134257480\">\n<p id=\"fs-id1165134257482\">Find 2<strong><em>u<\/em><\/strong> \u2212 3<strong><em>v<\/em><\/strong>.<\/p>\n<\/div>\n<div id=\"fs-id1165131835164\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131835167\">\u22124<strong><em>i<\/em><\/strong> \u2212 15<strong><em>j<\/em><\/strong><\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135673438\">\n<div id=\"fs-id1165135673439\">\n<p id=\"fs-id1165135673440\">Calculate<strong>[latex]\\,u\\cdot v.[\/latex]<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134060262\">\n<div id=\"fs-id1165134060263\">\n<p id=\"fs-id1165134060264\">Find a unit vector in the same direction as<strong>[latex]\\,v.[\/latex]<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165137772302\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137772304\">[latex]\\frac{2\\sqrt{13}}{13}i+\\frac{3\\sqrt{13}}{13}j[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132039481\">\n<div id=\"fs-id1165132039482\">\n<p id=\"fs-id1165132039483\">Given vector<strong>[latex]\\,v\\,[\/latex]<\/strong>has an initial point[latex]\\,{P}_{1}=\\left(2,2\\right)\\,[\/latex]and terminal point[latex]\\,{P}_{2}=\\left(-1,0\\right),\\,[\/latex]write the vector<strong>[latex]\\,v\\,[\/latex]<\/strong>in terms of<strong>[latex]\\,i\\,[\/latex]<\/strong>and<strong>[latex]\\,j.\\,[\/latex]<\/strong>On the graph, draw<strong>[latex]\\,v,\\,[\/latex]<\/strong>and<strong>[latex]\\,-v.\\,[\/latex]<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165133447852\">\n<dt>dot product<\/dt>\n<dd id=\"fs-id1165133447857\">given two vectors, the sum of the product of the horizontal components and the product of the vertical components<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133447862\">\n<dt>initial point<\/dt>\n<dd id=\"fs-id1165135369492\">the origin of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135369497\">\n<dt>magnitude<\/dt>\n<dd id=\"fs-id1165135369502\">the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135369506\">\n<dt>resultant<\/dt>\n<dd id=\"fs-id1165135369512\">a vector that results from addition or subtraction of two vectors, or from scalar multiplication<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238406\">\n<dt>scalar<\/dt>\n<dd id=\"fs-id1165135238411\">a quantity associated with magnitude but not direction; a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238414\">\n<dt>scalar multiplication<\/dt>\n<dd id=\"fs-id1165135238419\">the product of a constant and each component of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238423\">\n<dt>standard position<\/dt>\n<dd id=\"fs-id1165135369538\">the placement of a vector with the initial point at[latex]\\,\\left(0,0\\right)\\,[\/latex]and the terminal point[latex]\\,\\left(a,b\\right),\\,[\/latex]represented by the change in the <em>x<\/em>-coordinates and the change in the <em>y<\/em>-coordinates of the original vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134036770\">\n<dt>terminal point<\/dt>\n<dd id=\"fs-id1165133243502\">the end point of a vector, usually represented by an arrow indicating its direction<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133243505\">\n<dt>unit vector<\/dt>\n<dd id=\"fs-id1165133243510\">a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the <em>x<\/em>-axis and is defined as[latex]\\,{v}_{1}=\u23291,0\u232a\\,[\/latex]the vertical unit vector runs along the <em>y<\/em>-axis and is defined as[latex]\\,{v}_{2}=\u23290,1\u232a.[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131906701\">\n<dt>vector<\/dt>\n<dd id=\"fs-id1165131906706\">a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131906711\">\n<dt>vector addition<\/dt>\n<dd id=\"fs-id1165135700056\">the sum of two vectors, found by adding corresponding components<\/dd>\n<\/dl>\n<\/div>\n<\/div>\n","protected":false},"author":291,"menu_order":9,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-164","chapter","type-chapter","status-publish","hentry"],"part":147,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/164","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/164\/revisions"}],"predecessor-version":[{"id":165,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/164\/revisions\/165"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/147"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/164\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=164"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=164"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=164"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}