{"id":1584,"date":"2025-07-25T03:56:25","date_gmt":"2025-07-25T03:56:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1584"},"modified":"2026-03-25T04:02:01","modified_gmt":"2026-03-25T04:02:01","slug":"understanding-vectors-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/understanding-vectors-fresh-take\/","title":{"raw":"Vectors: Fresh Take","rendered":"Vectors: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>View vectors geometrically.<\/li>\r\n \t<li>Find magnitude and direction.<\/li>\r\n \t<li>Find the component form of a vector.<\/li>\r\n \t<li>Find the unit vector in the direction of v.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Vector Basics<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Some quantities need just a number\u2014like temperature or distance. These are <strong>scalars<\/strong>. But other quantities need both a number and a direction to make sense. For example, saying \"the wind is blowing at 15 mph\" is incomplete without knowing which direction. That's where <strong>vectors<\/strong> come in.<\/p>\r\n<p class=\"whitespace-normal break-words\">A <strong>vector<\/strong> has both <strong>magnitude<\/strong> (size or strength) and <strong>direction<\/strong>. We write them in bold like [latex]\\mathbf{v}[\/latex], or with an arrow when handwriting: [latex]\\overrightarrow{v}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Think of a vector as an arrow on a graph. It starts at the <strong>initial point<\/strong> and ends at the <strong>terminal point<\/strong>. The arrow's length shows magnitude ([latex]||\\mathbf{v}||[\/latex]), and the arrow points in the vector's direction.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Key Concepts<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Zero vector<\/strong> ([latex]\\mathbf{0}[\/latex]): Initial and terminal points are the same\u2014zero magnitude, no specific direction.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Equivalent vectors<\/strong>: Vectors with the same magnitude and direction are equal ([latex]\\mathbf{v} = \\mathbf{w}[\/latex]), even if they start at different points. Think of it like two people walking 5 miles north\u2014same journey, different starting locations.<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Quick Check for Equivalence<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Ask: Same length? Same direction? If yes to both, they're equivalent\u2014starting location doesn't matter.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Sketch the vector [latex]\\overrightarrow{ST}[\/latex] where [latex]S[\/latex] is point [latex](3,\u22121)[\/latex] and [latex]T[\/latex] is point [latex](\u22122,3)[\/latex].[reveal-answer q=\"fs-id1167793933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793933114\"][caption id=\"attachment_3476\" align=\"aligncenter\" width=\"417\"]<img class=\"size-full wp-image-3476\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28191727\/2-1-tryitans1.jpeg\" alt=\"This figure is a graph of the coordinate system. There is a line segment beginning at the ordered pair (3, -1). Also, this point is labeled \u201cS.\u201d The line segment ends at the ordered pair (-2, 3) and is labeled \u201cT.\u201d There is an arrowhead at point \u201cT,\u201d representing a vector. The line segment is labeled \u201cST.\u201d\" width=\"417\" height=\"347\" \/> Figure 3. The vector [latex]\\overrightarrow{ST}[\/latex].[\/caption][\/hidden-answer]<\/section>\r\n<h2>Vector Components<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Working with vectors geometrically (drawing arrows and parallelograms) works, but using coordinates makes calculations much easier and more precise.<\/p>\r\n<p class=\"whitespace-normal break-words\">Any vector can be written as [latex]\\mathbf{v} = \\langle x, y \\rangle[\/latex], where [latex]x[\/latex] and [latex]y[\/latex] are its <strong>components<\/strong>. This notation describes a vector starting at the origin [latex](0,0)[\/latex] and ending at [latex](x,y)[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Notation<\/strong>: Use angle brackets [latex]\\langle x, y \\rangle[\/latex] for vectors, not parentheses [latex](x,y)[\/latex]. Parentheses describe points; angle brackets describe vectors.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Converting to Component Form<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For a vector with initial point [latex](x_1, y_1)[\/latex] and terminal point [latex](x_2, y_2)[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{v} = \\langle x_2 - x_1, y_2 - y_1 \\rangle[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Think \"terminal minus initial\" for each coordinate. This gives you the horizontal and vertical change from start to finish.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Finding Magnitude<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The magnitude (length) of [latex]\\mathbf{v} = \\langle x, y \\rangle[\/latex] uses the distance formula:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]||\\mathbf{v}|| = \\sqrt{x^2 + y^2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This comes from the Pythagorean theorem\u2014the components form the legs of a right triangle, and the magnitude is the hypotenuse.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>From Magnitude and Direction to Components<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">If you know a vector's magnitude [latex]||\\mathbf{v}||[\/latex] and direction angle [latex]\\theta[\/latex] (measured from the positive [latex]x[\/latex]-axis), use trigonometry:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{v} = \\langle ||\\mathbf{v}|| \\cos \\theta, ||\\mathbf{v}|| \\sin \\theta \\rangle[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The cosine gives the horizontal component; the sine gives the vertical component.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Which of the following vectors are equivalent?\r\n\r\n[caption id=\"attachment_692\" align=\"aligncenter\" width=\"642\"]<img class=\"wp-image-692 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17190648\/2-1-12.jpeg\" alt=\"alt=&quot;This figure is a coordinate system with 6 vectors, each labeled a through f. Three of the vectors, \u201ca,\u201d \u201cb,\u201d and \u201ce\u201d have the same length and are pointing in the same direction.&quot;\" width=\"642\" height=\"384\" \/> Determine which vectors are equivalent.[\/caption]\r\n\r\n<div id=\"fs-id1167793940339\" class=\"exercise\">[reveal-answer q=\"fs-id1167794933124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794933124\"]\r\nVectors [latex]\\bf{a}[\/latex], [latex]\\bf{b}[\/latex], and [latex]\\bf{e}[\/latex] are equivalent.\r\n[\/hidden-answer]<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Vector [latex]\\bf{w}[\/latex] has initial point [latex](\u22124,\u22125)[\/latex] and terminal point [latex](\u22121,2)[\/latex]. Express [latex]\\bf{w}[\/latex] in component form.\r\n<div id=\"fs-id1167793940339\" class=\"exercise\">[reveal-answer q=\"fs-id1168794933124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1168794933124\"]\r\n[latex]\\langle 3,7 \\rangle[\/latex]\r\n[\/hidden-answer]<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Let [latex]\\bf{a}[\/latex][latex]=\\langle7,1\\rangle[\/latex] and let [latex]\\bf{b}[\/latex] be the vector with initial point [latex](3,2)[\/latex] and terminal point [latex](\u22121,\u22121)[\/latex].\r\n<ol style=\"list-style-type: lower-alpha;\">a. Find [latex]\\bf{||a||}[\/latex].<\/ol>\r\n<ol style=\"list-style-type: lower-alpha;\">b. Express [latex]\\bf{b}[\/latex] in component form.<\/ol>\r\n<ol style=\"list-style-type: lower-alpha;\">c. Find [latex]3[\/latex][latex]\\bf{a}[\/latex][latex]\u22124[\/latex][latex]\\bf{b}[\/latex].<\/ol>\r\n<div id=\"fs-id1167893940237\" class=\"exercise\">[reveal-answer q=\"fs-id1967793933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1967793933114\"]\r\n<ol style=\"list-style-type: lower-alpha;\">a. [latex]\\bf{||a||}[\/latex][latex] = 5\\sqrt{2}[\/latex]<\/ol>\r\n<ol style=\"list-style-type: lower-alpha;\">b. [latex]\\bf{b}[\/latex][latex] =\\langle-4,-3\\rangle[\/latex]<\/ol>\r\n<ol style=\"list-style-type: lower-alpha;\">c. [latex]3[\/latex][latex]\\bf{a}[\/latex][latex]\u22124[\/latex][latex]\\bf{b}[\/latex][latex] =\\langle37,15\\rangle[\/latex]<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the example above.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhaghfca-sA0145-1P3Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/sA0145-1P3Q?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hhaghfca-sA0145-1P3Q\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=7713415&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hhaghfca-sA0145-1P3Q&vembed=0&video_id=sA0145-1P3Q&video_target=tpm-plugin-hhaghfca-sA0145-1P3Q'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/CP+2.5_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCP 2.5\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the component form of vector [latex]{\\bf{v}}[\/latex] with magnitude [latex]10[\/latex] that forms an angle of [latex]120\u00b0[\/latex] with the positive [latex]x[\/latex]-axis.\r\n<div id=\"fs-id1167893944237\" class=\"exercise\">[reveal-answer q=\"fs-id1967793933284\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1967793933284\"]\r\n[latex]{\\bf{v}} = \\langle \u22125,5\\sqrt{3}\\rangle[\/latex]\r\n[\/hidden-answer]<\/div>\r\n<\/section>\r\n<h2>Unit Vectors<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">A <strong>unit vector<\/strong> has magnitude exactly [latex]1[\/latex]. Unit vectors are useful because they capture direction without being tied to any specific size\u2014think of them as \"direction indicators.\"<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Creating a Unit Vector (Normalization)<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">To find a unit vector [latex]\\mathbf{u}[\/latex] pointing in the same direction as any nonzero vector [latex]\\mathbf{v}[\/latex], divide [latex]\\mathbf{v}[\/latex] by its magnitude:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{u} = \\frac{1}{||\\mathbf{v}||}\\mathbf{v}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This process is called <strong>normalization<\/strong>. You're essentially scaling the vector down (or up) to length [latex]1[\/latex] while preserving its direction.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Why This Works<\/strong>: Dividing by [latex]||\\mathbf{v}||[\/latex] is just scalar multiplication by [latex]\\frac{1}{||\\mathbf{v}||[\/latex], which changes magnitude but not direction.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Building Vectors from Unit Vectors<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Once you have a unit vector [latex]\\mathbf{u}[\/latex] in the desired direction, you can create any vector in that direction by scalar multiplication. For example, [latex]7\\mathbf{u}[\/latex] has magnitude [latex]7[\/latex] and points the same way as [latex]\\mathbf{u}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Standard Unit Vectors [latex]\\mathbf{i}[\/latex] and [latex]\\mathbf{j}[\/latex]<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Two special unit vectors point along the coordinate axes:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\mathbf{i} = \\langle 1, 0 \\rangle[\/latex] (horizontal, along positive [latex]x[\/latex]-axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\mathbf{j} = \\langle 0, 1 \\rangle[\/latex] (vertical, along positive [latex]y[\/latex]-axis)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Linear Combination Form<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Any vector [latex]\\mathbf{v} = \\langle x, y \\rangle[\/latex] can be written using [latex]\\mathbf{i}[\/latex] and [latex]\\mathbf{j}[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{v} = x\\mathbf{i} + y\\mathbf{j}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This shows [latex]\\mathbf{v}[\/latex] as the sum of a horizontal component ([latex]x\\mathbf{i}[\/latex]) and a vertical component ([latex]y\\mathbf{j}[\/latex]).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Unit Vectors from Angles<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">If a unit vector makes angle [latex]\\theta[\/latex] with the positive [latex]x[\/latex]-axis, its components come directly from the unit circle:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{u} = \\langle \\cos \\theta, \\sin \\theta \\rangle = (\\cos \\theta)\\mathbf{i} + (\\sin \\theta)\\mathbf{j}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]{\\bf{v}} = \\langle 9,2 \\rangle[\/latex]. Find a vector with magnitude [latex]5[\/latex] in the opposite direction as [latex]{\\bf{v}}[\/latex].\r\n<div id=\"fs-id1167894444237\" class=\"exercise\">[reveal-answer q=\"fs-id1967793921284\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1967793921284\"]\r\n[latex] \\langle -\\frac{45}{\\sqrt{85}}, -\\frac{10}{\\sqrt{85}}\\rangle[\/latex]\r\n[\/hidden-answer]<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Let [latex]{\\bf{a}} = \\langle 16,-11 \\rangle [\/latex] and let [latex]{\\bf{b}}[\/latex] be a unit vector that forms an angle of [latex]225\u00b0[\/latex] with the positive [latex]x[\/latex]-axis. Express [latex]{\\bf{a}}[\/latex] and [latex]{\\bf{b}}[\/latex] in terms of the standard unit vectors.\r\n<div id=\"fs-id1167895644237\" class=\"exercise\">[reveal-answer q=\"fs-id1968393921284\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1968393921284\"]\r\n[latex]{\\bf{a}} = 16{\\bf{i}} - 11{\\bf{j}}[\/latex]\r\n[latex]{\\bf{b}} = -\\frac{\\sqrt{2}}{2}{\\bf{i}} - \\frac{\\sqrt{2}}{2}{\\bf{j}}[\/latex]\r\n[\/hidden-answer]<\/div>\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above Try IT.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gcbeccfc-azDVrpxgUko\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/azDVrpxgUko?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gcbeccfc-azDVrpxgUko\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=7713416&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gcbeccfc-azDVrpxgUko&vembed=0&video_id=azDVrpxgUko&video_target=tpm-plugin-gcbeccfc-azDVrpxgUko'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/CP+2.9_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCP 2.9\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>View vectors geometrically.<\/li>\n<li>Find magnitude and direction.<\/li>\n<li>Find the component form of a vector.<\/li>\n<li>Find the unit vector in the direction of v.<\/li>\n<\/ul>\n<\/section>\n<h2>Vector Basics<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Some quantities need just a number\u2014like temperature or distance. These are <strong>scalars<\/strong>. But other quantities need both a number and a direction to make sense. For example, saying &#8220;the wind is blowing at 15 mph&#8221; is incomplete without knowing which direction. That&#8217;s where <strong>vectors<\/strong> come in.<\/p>\n<p class=\"whitespace-normal break-words\">A <strong>vector<\/strong> has both <strong>magnitude<\/strong> (size or strength) and <strong>direction<\/strong>. We write them in bold like [latex]\\mathbf{v}[\/latex], or with an arrow when handwriting: [latex]\\overrightarrow{v}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Think of a vector as an arrow on a graph. It starts at the <strong>initial point<\/strong> and ends at the <strong>terminal point<\/strong>. The arrow&#8217;s length shows magnitude ([latex]||\\mathbf{v}||[\/latex]), and the arrow points in the vector&#8217;s direction.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Key Concepts<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Zero vector<\/strong> ([latex]\\mathbf{0}[\/latex]): Initial and terminal points are the same\u2014zero magnitude, no specific direction.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Equivalent vectors<\/strong>: Vectors with the same magnitude and direction are equal ([latex]\\mathbf{v} = \\mathbf{w}[\/latex]), even if they start at different points. Think of it like two people walking 5 miles north\u2014same journey, different starting locations.<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Quick Check for Equivalence<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Ask: Same length? Same direction? If yes to both, they&#8217;re equivalent\u2014starting location doesn&#8217;t matter.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch the vector [latex]\\overrightarrow{ST}[\/latex] where [latex]S[\/latex] is point [latex](3,\u22121)[\/latex] and [latex]T[\/latex] is point [latex](\u22122,3)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793933114\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793933114\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_3476\" aria-describedby=\"caption-attachment-3476\" style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-3476\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28191727\/2-1-tryitans1.jpeg\" alt=\"This figure is a graph of the coordinate system. There is a line segment beginning at the ordered pair (3, -1). Also, this point is labeled \u201cS.\u201d The line segment ends at the ordered pair (-2, 3) and is labeled \u201cT.\u201d There is an arrowhead at point \u201cT,\u201d representing a vector. The line segment is labeled \u201cST.\u201d\" width=\"417\" height=\"347\" \/><figcaption id=\"caption-attachment-3476\" class=\"wp-caption-text\">Figure 3. The vector [latex]\\overrightarrow{ST}[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<h2>Vector Components<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Working with vectors geometrically (drawing arrows and parallelograms) works, but using coordinates makes calculations much easier and more precise.<\/p>\n<p class=\"whitespace-normal break-words\">Any vector can be written as [latex]\\mathbf{v} = \\langle x, y \\rangle[\/latex], where [latex]x[\/latex] and [latex]y[\/latex] are its <strong>components<\/strong>. This notation describes a vector starting at the origin [latex](0,0)[\/latex] and ending at [latex](x,y)[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Notation<\/strong>: Use angle brackets [latex]\\langle x, y \\rangle[\/latex] for vectors, not parentheses [latex](x,y)[\/latex]. Parentheses describe points; angle brackets describe vectors.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Converting to Component Form<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For a vector with initial point [latex](x_1, y_1)[\/latex] and terminal point [latex](x_2, y_2)[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{v} = \\langle x_2 - x_1, y_2 - y_1 \\rangle[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Think &#8220;terminal minus initial&#8221; for each coordinate. This gives you the horizontal and vertical change from start to finish.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Finding Magnitude<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The magnitude (length) of [latex]\\mathbf{v} = \\langle x, y \\rangle[\/latex] uses the distance formula:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]||\\mathbf{v}|| = \\sqrt{x^2 + y^2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This comes from the Pythagorean theorem\u2014the components form the legs of a right triangle, and the magnitude is the hypotenuse.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>From Magnitude and Direction to Components<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">If you know a vector&#8217;s magnitude [latex]||\\mathbf{v}||[\/latex] and direction angle [latex]\\theta[\/latex] (measured from the positive [latex]x[\/latex]-axis), use trigonometry:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{v} = \\langle ||\\mathbf{v}|| \\cos \\theta, ||\\mathbf{v}|| \\sin \\theta \\rangle[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The cosine gives the horizontal component; the sine gives the vertical component.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Which of the following vectors are equivalent?<\/p>\n<figure id=\"attachment_692\" aria-describedby=\"caption-attachment-692\" style=\"width: 642px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-692 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17190648\/2-1-12.jpeg\" alt=\"alt=&quot;This figure is a coordinate system with 6 vectors, each labeled a through f. Three of the vectors, \u201ca,\u201d \u201cb,\u201d and \u201ce\u201d have the same length and are pointing in the same direction.&quot;\" width=\"642\" height=\"384\" \/><figcaption id=\"caption-attachment-692\" class=\"wp-caption-text\">Determine which vectors are equivalent.<\/figcaption><\/figure>\n<div id=\"fs-id1167793940339\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794933124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794933124\" class=\"hidden-answer\" style=\"display: none\">\nVectors [latex]\\bf{a}[\/latex], [latex]\\bf{b}[\/latex], and [latex]\\bf{e}[\/latex] are equivalent.\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Vector [latex]\\bf{w}[\/latex] has initial point [latex](\u22124,\u22125)[\/latex] and terminal point [latex](\u22121,2)[\/latex]. Express [latex]\\bf{w}[\/latex] in component form.<\/p>\n<div class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1168794933124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1168794933124\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\langle 3,7 \\rangle[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]\\bf{a}[\/latex][latex]=\\langle7,1\\rangle[\/latex] and let [latex]\\bf{b}[\/latex] be the vector with initial point [latex](3,2)[\/latex] and terminal point [latex](\u22121,\u22121)[\/latex].<\/p>\n<ol style=\"list-style-type: lower-alpha;\">  <\/ol>\n<ol style=\"list-style-type: lower-alpha;\">     <\/ol>\n<ol style=\"list-style-type: lower-alpha;\">  <\/ol>\n<div id=\"fs-id1167893940237\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1967793933114\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1967793933114\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">   <\/ol>\n<ol style=\"list-style-type: lower-alpha;\">  <\/ol>\n<ol style=\"list-style-type: lower-alpha;\">  <\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the example above.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhaghfca-sA0145-1P3Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/sA0145-1P3Q?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hhaghfca-sA0145-1P3Q\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=7713415&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hhaghfca-sA0145-1P3Q&#38;vembed=0&#38;video_id=sA0145-1P3Q&#38;video_target=tpm-plugin-hhaghfca-sA0145-1P3Q\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/CP+2.5_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCP 2.5\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the component form of vector [latex]{\\bf{v}}[\/latex] with magnitude [latex]10[\/latex] that forms an angle of [latex]120\u00b0[\/latex] with the positive [latex]x[\/latex]-axis.<\/p>\n<div id=\"fs-id1167893944237\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1967793933284\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1967793933284\" class=\"hidden-answer\" style=\"display: none\">\n[latex]{\\bf{v}} = \\langle \u22125,5\\sqrt{3}\\rangle[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Unit Vectors<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A <strong>unit vector<\/strong> has magnitude exactly [latex]1[\/latex]. Unit vectors are useful because they capture direction without being tied to any specific size\u2014think of them as &#8220;direction indicators.&#8221;<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Creating a Unit Vector (Normalization)<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">To find a unit vector [latex]\\mathbf{u}[\/latex] pointing in the same direction as any nonzero vector [latex]\\mathbf{v}[\/latex], divide [latex]\\mathbf{v}[\/latex] by its magnitude:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{u} = \\frac{1}{||\\mathbf{v}||}\\mathbf{v}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This process is called <strong>normalization<\/strong>. You&#8217;re essentially scaling the vector down (or up) to length [latex]1[\/latex] while preserving its direction.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Why This Works<\/strong>: Dividing by [latex]||\\mathbf{v}||[\/latex] is just scalar multiplication by [latex]\\frac{1}{||\\mathbf{v}||[\/latex], which changes magnitude but not direction.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Building Vectors from Unit Vectors<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Once you have a unit vector [latex]\\mathbf{u}[\/latex] in the desired direction, you can create any vector in that direction by scalar multiplication. For example, [latex]7\\mathbf{u}[\/latex] has magnitude [latex]7[\/latex] and points the same way as [latex]\\mathbf{u}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Standard Unit Vectors [latex]\\mathbf{i}[\/latex] and [latex]\\mathbf{j}[\/latex]<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Two special unit vectors point along the coordinate axes:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\mathbf{i} = \\langle 1, 0 \\rangle[\/latex] (horizontal, along positive [latex]x[\/latex]-axis)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\mathbf{j} = \\langle 0, 1 \\rangle[\/latex] (vertical, along positive [latex]y[\/latex]-axis)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Linear Combination Form<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Any vector [latex]\\mathbf{v} = \\langle x, y \\rangle[\/latex] can be written using [latex]\\mathbf{i}[\/latex] and [latex]\\mathbf{j}[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{v} = x\\mathbf{i} + y\\mathbf{j}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This shows [latex]\\mathbf{v}[\/latex] as the sum of a horizontal component ([latex]x\\mathbf{i}[\/latex]) and a vertical component ([latex]y\\mathbf{j}[\/latex]).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Unit Vectors from Angles<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">If a unit vector makes angle [latex]\\theta[\/latex] with the positive [latex]x[\/latex]-axis, its components come directly from the unit circle:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{u} = \\langle \\cos \\theta, \\sin \\theta \\rangle = (\\cos \\theta)\\mathbf{i} + (\\sin \\theta)\\mathbf{j}[\/latex]<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]{\\bf{v}} = \\langle 9,2 \\rangle[\/latex]. Find a vector with magnitude [latex]5[\/latex] in the opposite direction as [latex]{\\bf{v}}[\/latex].<\/p>\n<div id=\"fs-id1167894444237\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1967793921284\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1967793921284\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\langle -\\frac{45}{\\sqrt{85}}, -\\frac{10}{\\sqrt{85}}\\rangle[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]{\\bf{a}} = \\langle 16,-11 \\rangle[\/latex] and let [latex]{\\bf{b}}[\/latex] be a unit vector that forms an angle of [latex]225\u00b0[\/latex] with the positive [latex]x[\/latex]-axis. Express [latex]{\\bf{a}}[\/latex] and [latex]{\\bf{b}}[\/latex] in terms of the standard unit vectors.<\/p>\n<div id=\"fs-id1167895644237\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1968393921284\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1968393921284\" class=\"hidden-answer\" style=\"display: none\">\n[latex]{\\bf{a}} = 16{\\bf{i}} - 11{\\bf{j}}[\/latex]<br \/>\n[latex]{\\bf{b}} = -\\frac{\\sqrt{2}}{2}{\\bf{i}} - \\frac{\\sqrt{2}}{2}{\\bf{j}}[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above Try IT.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gcbeccfc-azDVrpxgUko\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/azDVrpxgUko?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gcbeccfc-azDVrpxgUko\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=7713416&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gcbeccfc-azDVrpxgUko&#38;vembed=0&#38;video_id=azDVrpxgUko&#38;video_target=tpm-plugin-gcbeccfc-azDVrpxgUko\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/CP+2.9_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCP 2.9\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"CP 2.5\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/sA0145-1P3Q\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"CP 2.9\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/azDVrpxgUko\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":520,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"CP 2.5","author":"Ryan Melton","organization":"","url":"https:\/\/youtu.be\/sA0145-1P3Q","project":"","license":"arr","license_terms":"Standard 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