{"id":492,"date":"2025-10-09T15:14:56","date_gmt":"2025-10-09T15:14:56","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=492"},"modified":"2025-10-24T12:40:04","modified_gmt":"2025-10-24T12:40:04","slug":"vector-basics-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/vector-basics-fresh-take\/","title":{"raw":"Vector Basics: Fresh Take","rendered":"Vector Basics: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Explain what vectors are and how to write them using proper notation, with real-world examples of vector quantities<\/li>\r\n \t<li>Perform basic vector operations (scalar multiplication, addition, subtraction)<\/li>\r\n \t<li>Write vectors using their components and find how long they are (their magnitude)<\/li>\r\n \t<li>Express a vector in terms of unit vectors<\/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>Combining 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\">Vectors show up constantly in real life\u2014think about a boat crossing a river (engine pushing one way, current pushing another) or a quarterback throwing a football (force and angle combined). To solve these problems, you need to know how to combine vectors.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Scalar Multiplication: Stretching and Flipping<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">When you multiply a vector [latex]\\mathbf{v}[\/latex] by a scalar [latex]k[\/latex], you change its length and possibly its direction:<\/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>If [latex]k &gt; 0[\/latex]<\/strong>: The vector stretches or shrinks but keeps the same direction. For example, [latex]2\\mathbf{v}[\/latex] doubles the length; [latex]0.5\\mathbf{v}[\/latex] halves it.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>If [latex]k &lt; 0[\/latex]<\/strong>: The vector flips to point the opposite direction. For instance, [latex]-\\mathbf{v}[\/latex] has the same length as [latex]\\mathbf{v}[\/latex] but points the other way.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>If [latex]k = 0[\/latex]<\/strong>: You get the zero vector [latex]\\mathbf{0}[\/latex].<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Vector Addition: Two Methods, Same Result<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">To add vectors [latex]\\mathbf{v} + \\mathbf{w}[\/latex], use either approach:<\/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>Triangle Method<\/strong>: Place the starting point of [latex]\\mathbf{w}[\/latex] at the endpoint of [latex]\\mathbf{v}[\/latex]. The sum runs from [latex]\\mathbf{v}[\/latex]'s start to [latex]\\mathbf{w}[\/latex]'s end. Think of it like walking from A to B, then B to C\u2014your total displacement goes directly from A to C.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parallelogram Method<\/strong>: Start both vectors at the same point and complete the parallelogram. The diagonal is your sum.<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Order doesn't matter: [latex]\\mathbf{v} + \\mathbf{w} = \\mathbf{w} + \\mathbf{v}[\/latex] (commutative property).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Triangle Inequality<\/strong>: The sum's length satisfies [latex]||\\mathbf{v} + \\mathbf{w}|| \\leq ||\\mathbf{v}|| + ||\\mathbf{w}||[\/latex]. Equality only happens when vectors point the same direction.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Vector Subtraction: Adding the Opposite<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]\\mathbf{v} - \\mathbf{w}[\/latex], rewrite it as [latex]\\mathbf{v} + (-\\mathbf{w})[\/latex]. Graphically, the difference points from [latex]\\mathbf{w}[\/latex]'s endpoint to [latex]\\mathbf{v}[\/latex]'s endpoint.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Using vectors [latex]\\bf{w}[\/latex] and [latex]\\bf{w}[\/latex] from Example: Combining Vectors, sketch the vector [latex]\\bf{2w - v}[\/latex].\r\n<div id=\"fs-id1167793940239\" class=\"exercise\">[reveal-answer q=\"fs-id1167793933124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793933124\"][caption id=\"attachment_3477\" align=\"aligncenter\" width=\"270\"]<img class=\"size-full wp-image-3477\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28191840\/2-1-tryitans2.jpeg\" alt=\"This figure has two vectors. They are vector v and vector w. They are not connected.\" width=\"270\" height=\"271\" \/> Figure 11. The graph of the combined vector [latex]\\bf{2w - v}[\/latex].[\/caption][\/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 example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7699685&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=0MXVlIKjB4Q&amp;video_target=tpm-plugin-3159nf61-0MXVlIKjB4Q\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.2_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCP 2.2\u201d here (opens in new window).<\/a><\/p>\r\n\r\n<\/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\"><strong>Component Form: The Key Idea<\/strong><\/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>Operations in Component Form<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Component form makes calculations algebraic instead of geometric:<\/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>Scalar multiplication<\/strong>: [latex]k\\mathbf{v} = \\langle kx, ky \\rangle[\/latex] (multiply each component by [latex]k[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Vector addition<\/strong>: [latex]\\mathbf{v} + \\mathbf{w} = \\langle x_1 + x_2, y_1 + y_2 \\rangle[\/latex] (add corresponding components)<\/li>\r\n<\/ul>\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\" \/> Figure 3. 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.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7713415&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=sA0145-1P3Q&amp;video_target=tpm-plugin-7hhonzao-sA0145-1P3Q\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.5_transcript.html\">transcript for \u201cCP 2.5\u201d here (opens in new window)<\/a><\/center><\/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.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7713416&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=azDVrpxgUko&amp;video_target=tpm-plugin-ihc7z8ph-azDVrpxgUko\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.9_transcript.html\">transcript for \u201cCP 2.9\u201d here (opens in new window)<\/a><\/center><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Explain what vectors are and how to write them using proper notation, with real-world examples of vector quantities<\/li>\n<li>Perform basic vector operations (scalar multiplication, addition, subtraction)<\/li>\n<li>Write vectors using their components and find how long they are (their magnitude)<\/li>\n<li>Express a vector in terms of unit vectors<\/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>Combining Vectors<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Vectors show up constantly in real life\u2014think about a boat crossing a river (engine pushing one way, current pushing another) or a quarterback throwing a football (force and angle combined). To solve these problems, you need to know how to combine vectors.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Scalar Multiplication: Stretching and Flipping<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When you multiply a vector [latex]\\mathbf{v}[\/latex] by a scalar [latex]k[\/latex], you change its length and possibly its direction:<\/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>If [latex]k > 0[\/latex]<\/strong>: The vector stretches or shrinks but keeps the same direction. For example, [latex]2\\mathbf{v}[\/latex] doubles the length; [latex]0.5\\mathbf{v}[\/latex] halves it.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>If [latex]k < 0[\/latex]<\/strong>: The vector flips to point the opposite direction. For instance, [latex]-\\mathbf{v}[\/latex] has the same length as [latex]\\mathbf{v}[\/latex] but points the other way.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>If [latex]k = 0[\/latex]<\/strong>: You get the zero vector [latex]\\mathbf{0}[\/latex].<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Vector Addition: Two Methods, Same Result<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">To add vectors [latex]\\mathbf{v} + \\mathbf{w}[\/latex], use either approach:<\/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>Triangle Method<\/strong>: Place the starting point of [latex]\\mathbf{w}[\/latex] at the endpoint of [latex]\\mathbf{v}[\/latex]. The sum runs from [latex]\\mathbf{v}[\/latex]&#8216;s start to [latex]\\mathbf{w}[\/latex]&#8216;s end. Think of it like walking from A to B, then B to C\u2014your total displacement goes directly from A to C.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Parallelogram Method<\/strong>: Start both vectors at the same point and complete the parallelogram. The diagonal is your sum.<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Order doesn&#8217;t matter: [latex]\\mathbf{v} + \\mathbf{w} = \\mathbf{w} + \\mathbf{v}[\/latex] (commutative property).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Triangle Inequality<\/strong>: The sum&#8217;s length satisfies [latex]||\\mathbf{v} + \\mathbf{w}|| \\leq ||\\mathbf{v}|| + ||\\mathbf{w}||[\/latex]. Equality only happens when vectors point the same direction.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Vector Subtraction: Adding the Opposite<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For [latex]\\mathbf{v} - \\mathbf{w}[\/latex], rewrite it as [latex]\\mathbf{v} + (-\\mathbf{w})[\/latex]. Graphically, the difference points from [latex]\\mathbf{w}[\/latex]&#8216;s endpoint to [latex]\\mathbf{v}[\/latex]&#8216;s endpoint.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Using vectors [latex]\\bf{w}[\/latex] and [latex]\\bf{w}[\/latex] from Example: Combining Vectors, sketch the vector [latex]\\bf{2w - v}[\/latex].<\/p>\n<div id=\"fs-id1167793940239\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793933124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793933124\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_3477\" aria-describedby=\"caption-attachment-3477\" style=\"width: 270px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-3477\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28191840\/2-1-tryitans2.jpeg\" alt=\"This figure has two vectors. They are vector v and vector w. They are not connected.\" width=\"270\" height=\"271\" \/><figcaption id=\"caption-attachment-3477\" class=\"wp-caption-text\">Figure 11. The graph of the combined vector [latex]\\bf{2w - v}[\/latex].<\/figcaption><\/figure>\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 example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7699685&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=0MXVlIKjB4Q&amp;video_target=tpm-plugin-3159nf61-0MXVlIKjB4Q\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.2_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCP 2.2\u201d here (opens in new window).<\/a><\/p>\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\"><strong>Component Form: The Key Idea<\/strong><\/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>Operations in Component Form<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Component form makes calculations algebraic instead of geometric:<\/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>Scalar multiplication<\/strong>: [latex]k\\mathbf{v} = \\langle kx, ky \\rangle[\/latex] (multiply each component by [latex]k[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Vector addition<\/strong>: [latex]\\mathbf{v} + \\mathbf{w} = \\langle x_1 + x_2, y_1 + y_2 \\rangle[\/latex] (add corresponding components)<\/li>\n<\/ul>\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\">Figure 3. 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.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7713415&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=sA0145-1P3Q&amp;video_target=tpm-plugin-7hhonzao-sA0145-1P3Q\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.5_transcript.html\">transcript for \u201cCP 2.5\u201d here (opens in new window)<\/a><\/div>\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.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7713416&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=azDVrpxgUko&amp;video_target=tpm-plugin-ihc7z8ph-azDVrpxgUko\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.9_transcript.html\">transcript for \u201cCP 2.9\u201d here (opens in new window)<\/a><\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-492","chapter","type-chapter","status-publish","hentry"],"part":420,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/492","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/492\/revisions"}],"predecessor-version":[{"id":1741,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/492\/revisions\/1741"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/420"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/492\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=492"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=492"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=492"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=492"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}