{"id":552,"date":"2025-10-09T15:20:46","date_gmt":"2025-10-09T15:20:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=552"},"modified":"2025-12-02T14:43:43","modified_gmt":"2025-12-02T14:43:43","slug":"introduction-to-vector-valued-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-vector-valued-functions-fresh-take\/","title":{"raw":"Introduction to Vector-Valued Functions: Fresh Take","rendered":"Introduction to Vector-Valued Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write and understand vector-valued functions using components and unit vectors<\/li>\r\n \t<li>Recognize parametric equations for a space curve<\/li>\r\n \t<li>Define the limit of a vector-valued function<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Vector-Valued Functions<\/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\"><strong>Vector-valued functions<\/strong> output vectors instead of numbers, allowing us to describe curves and motion in 2D and 3D space.<\/p>\r\n<p class=\"whitespace-normal break-words\">A vector-valued function takes parameter [latex]t[\/latex] and produces a vector:<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>2D<\/strong>: [latex]\\mathbf{r}(t) = \\langle f(t), g(t) \\rangle[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>3D<\/strong>: [latex]\\mathbf{r}(t) = \\langle f(t), g(t), h(t) \\rangle[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The functions [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are <strong>component functions<\/strong>. Each [latex]t[\/latex]-value produces one vector (point in space). As [latex]t[\/latex] varies, the vectors trace out a curve.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Domain<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The domain is the <strong>intersection<\/strong> of all component function domains. Exclude values where any component is undefined.<\/p>\r\n<p class=\"whitespace-normal break-words\">Process:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Find each component's domain individually<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Take the intersection of these domains<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Exclude values where any component is undefined<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Common scenarios<\/strong>: Polynomials (all real numbers), bounded intervals ([latex]a \\leq t \\leq b[\/latex]), or restrictions from [latex]\\tan t[\/latex], [latex]\\ln t[\/latex], [latex]\\sqrt{t}[\/latex], or [latex]\\frac{1}{f(t)}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Range<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Vector-valued functions output vectors, not numbers:<\/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>2D functions<\/strong>: Range is a subset of [latex]\\mathbb{R}^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>3D functions<\/strong>: Range is a subset of [latex]\\mathbb{R}^3[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Geometric Interpretation<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Vector-valued functions describe <strong>parametric curves<\/strong>. As [latex]t[\/latex] varies, the tip of [latex]\\mathbf{r}(t)[\/latex] traces out a curve in space.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Example<\/strong>: [latex]\\mathbf{r}(t) = 4\\cos t \\mathbf{i} + 3\\sin t \\mathbf{j}[\/latex] traces an ellipse as [latex]t[\/latex] goes from [latex]0[\/latex] to [latex]2\\pi[\/latex].<\/p>\r\n\r\n<\/div>\r\n<h2>Graphing Vector-Valued Functions<\/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\">When you graph a vector-valued function, you're essentially plotting a curve by tracking where vectors point as the parameter [latex]t[\/latex] changes. The key difference from regular parametric curves? You're visualizing vectors in <strong>standard position<\/strong> (starting at the origin) rather than just points.<\/p>\r\n<p class=\"whitespace-normal break-words\">Think of it this way: as [latex]t[\/latex] increases, each vector [latex]{\\bf{r}}(t)[\/latex] points from the origin to a specific location in space. The <strong>terminal points<\/strong> (where these vectors end) trace out your curve. The curve's shape depends on your function's form:<\/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>Plane curve<\/strong>: [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}}[\/latex] creates a curve in the [latex]xy[\/latex]-plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Space curve<\/strong>: [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}} + h(t){\\bf{k}}[\/latex] creates a curve in three-dimensional space<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Graphing Strategy<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Make a table<\/strong>: Pick several [latex]t[\/latex]-values and calculate [latex]{\\bf{r}}(t)[\/latex] for each<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Draw vectors<\/strong>: Place each vector with its tail at the origin<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Mark endpoints<\/strong>: Note where each vector's tip lands<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Connect smoothly<\/strong>: Draw a curve through these terminal points in order<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Show direction<\/strong>: Add arrows indicating how the curve progresses as [latex]t[\/latex] increases<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">Vector-valued functions and parametric equations are two ways of describing the same curve. Given [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}}[\/latex], you can write [latex]x = f(t)[\/latex] and [latex]y = g(t)[\/latex] to get the parametric form\u2014the graphs match exactly.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Reparameterization<\/strong> means different functions can trace the same curve. For example, [latex]{\\bf{r}}(t) = 4\\cos t{\\bf{i}} + 3\\sin t{\\bf{j}}[\/latex] and [latex]{\\bf{r}}(t) = 4\\cos t^3{\\bf{i}} + 3\\sin t^3{\\bf{j}}[\/latex] both produce ellipses\u2014the shape stays the same even though the parameter changes. Only the speed at which you trace the curve differs.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Special Case: Helices<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A <strong>helix<\/strong> is a three-dimensional spiral that looks like a spring. It has a circular or elliptical cross-section when viewed from above and climbs (or descends) as [latex]t[\/latex] increases. If you remove the [latex]{\\bf{k}[\/latex] component, you're left with the curve's projection onto the [latex]xy[\/latex]-plane.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Create a graph of the vector valued function [latex]{\\bf{r}}\\,(t)=(t^{2}-1)\\,{\\bf{i}}+(2t-3)\\,{\\bf{j}},\\ 0\\ \\leq\\ t\\ \\leq\\ 3.[\/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\"]<\/div>\r\n<div>[caption id=\"attachment_5650\" align=\"aligncenter\" width=\"417\"]<img class=\"size-full wp-image-5650\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2022\/04\/03163833\/Checkpoint-3.2.jpeg\" alt=\"This figure is a graph of the function r(t) = (t^2-1)i + (2t-3)j, for the values of t from 0 to 3. The curve begins in the 3rd quadrant at the ordered pair (-1,-3) and increases up through the 1st quadrant. It is increasing and has arrows on the curve representing orientation to the right.\" width=\"417\" height=\"347\" \/> Figure 4. Graph of [latex]{\\bf{r}}\\,(t)=(t^{2}-1)\\,{\\bf{i}}+(2t-3)\\,{\\bf{j}},\\ 0\\ \\leq\\ t\\ \\leq\\ 3[\/latex].[\/caption]<\/div>\r\n<div class=\"exercise\">[\/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=7949574&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=nRScwQ-_TBU&amp;video_target=tpm-plugin-umi22qm5-nRScwQ-_TBU\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP3.2_transcript.html\">transcript for \u201cCP 3.2\u201d here (opens in new window).<\/a><\/center><\/section>\r\n<h2>Limits and Continuity of a Vector-Valued Function<\/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\">Finding limits and checking continuity for vector-valued functions works exactly like it does for regular functions\u2014you just handle each component separately.<\/p>\r\n<p class=\"whitespace-normal break-words\">Instead of wrestling with the formal definition involving [latex]|{\\bf{r}}(t) - {\\bf{L}}|[\/latex], use this practical method: <strong>take the limit of each component function individually<\/strong>.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>For plane curves<\/strong> [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}}[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\underset{t\\to a}{\\text{lim}}{\\bf{r}}(t)=\\bigg[\\underset{t\\to a}{\\text{lim}}f(t)\\bigg]{\\bf{i}}+\\bigg[\\underset{t\\to a}{\\text{lim}}g(t)\\bigg]{\\bf{j}}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>For space curves<\/strong> [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}} + h(t){\\bf{k}}[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\underset{t\\to a}{\\text{lim}}{\\bf{r}}(t)=\\bigg[\\underset{t\\to a}{\\text{lim}}f(t)\\bigg]{\\bf{i}}+\\bigg[\\underset{t\\to a}{\\text{lim}}g(t)\\bigg]{\\bf{j}}+\\bigg[\\underset{t\\to a}{\\text{lim}}h(t)\\bigg]{\\bf{k}}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This only works if <strong>all<\/strong> the component limits exist.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>How to Calculate Limits<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Break the vector function into its parts ([latex]{\\bf{i}}[\/latex], [latex]{\\bf{j}}[\/latex], and [latex]{\\bf{k}}[\/latex] components), find the limit of each scalar function using your usual techniques (substitution, algebraic manipulation, etc.), then reassemble the results into vector form. Each component follows the same limit rules you already know.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Checking Continuity<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A vector-valued function [latex]{\\bf{r}}(t)[\/latex] is continuous at [latex]t = a[\/latex] when these three familiar conditions hold:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]{\\bf{r}}(a)[\/latex] exists (the function is defined at that point)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\lim_{t\\to a}{\\bf{r}}(t)[\/latex] exists (the limit exists as you approach the point)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\lim_{t\\to a}{\\bf{r}}(t) = {\\bf{r}}(a)[\/latex] (the limit equals the function value)<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">These are the same three requirements you learned for regular functions. A vector-valued function is continuous at a point when its graph flows smoothly through that point without breaks, jumps, or gaps\u2014you could trace it without lifting your pencil.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Calculate [latex]\\displaystyle\\lim_{t{\\to}-2}{\\bf{r}}\\,(t)[\/latex] for the function [latex]{\\bf{r}}\\,(t)=\\sqrt{t^{2}-3t-1}\\,{\\bf{i}}+(4t+3)\\,{\\bf{j}}+\\sin{\\frac{(t+1)\\,\\pi}{2}}\\,{\\bf{k}}[\/latex].\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\n<div>[latex]\\displaystyle\\lim_{t{\\to}-2}{\\bf{r}}\\,(t)=3\\,{\\bf{i}}-5\\,{\\bf{j}}-{\\bf{k}}[\/latex]<\/div>\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 above Try It<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7949584&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=UkThcL4VPfc&amp;video_target=tpm-plugin-x7y3bb4i-UkThcL4VPfc\" 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\/CP3.3_transcript.html\">transcript for \u201cCP 3.3\u201d here (opens in new window).<\/a><\/center><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write and understand vector-valued functions using components and unit vectors<\/li>\n<li>Recognize parametric equations for a space curve<\/li>\n<li>Define the limit of a vector-valued function<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Vector-Valued Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Vector-valued functions<\/strong> output vectors instead of numbers, allowing us to describe curves and motion in 2D and 3D space.<\/p>\n<p class=\"whitespace-normal break-words\">A vector-valued function takes parameter [latex]t[\/latex] and produces a vector:<\/p>\n<p class=\"whitespace-normal break-words\"><strong>2D<\/strong>: [latex]\\mathbf{r}(t) = \\langle f(t), g(t) \\rangle[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>3D<\/strong>: [latex]\\mathbf{r}(t) = \\langle f(t), g(t), h(t) \\rangle[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The functions [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are <strong>component functions<\/strong>. Each [latex]t[\/latex]-value produces one vector (point in space). As [latex]t[\/latex] varies, the vectors trace out a curve.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Domain<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The domain is the <strong>intersection<\/strong> of all component function domains. Exclude values where any component is undefined.<\/p>\n<p class=\"whitespace-normal break-words\">Process:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Find each component&#8217;s domain individually<\/li>\n<li class=\"whitespace-normal break-words\">Take the intersection of these domains<\/li>\n<li class=\"whitespace-normal break-words\">Exclude values where any component is undefined<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Common scenarios<\/strong>: Polynomials (all real numbers), bounded intervals ([latex]a \\leq t \\leq b[\/latex]), or restrictions from [latex]\\tan t[\/latex], [latex]\\ln t[\/latex], [latex]\\sqrt{t}[\/latex], or [latex]\\frac{1}{f(t)}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Range<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Vector-valued functions output vectors, not numbers:<\/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>2D functions<\/strong>: Range is a subset of [latex]\\mathbb{R}^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>3D functions<\/strong>: Range is a subset of [latex]\\mathbb{R}^3[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Geometric Interpretation<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Vector-valued functions describe <strong>parametric curves<\/strong>. As [latex]t[\/latex] varies, the tip of [latex]\\mathbf{r}(t)[\/latex] traces out a curve in space.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Example<\/strong>: [latex]\\mathbf{r}(t) = 4\\cos t \\mathbf{i} + 3\\sin t \\mathbf{j}[\/latex] traces an ellipse as [latex]t[\/latex] goes from [latex]0[\/latex] to [latex]2\\pi[\/latex].<\/p>\n<\/div>\n<h2>Graphing Vector-Valued Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When you graph a vector-valued function, you&#8217;re essentially plotting a curve by tracking where vectors point as the parameter [latex]t[\/latex] changes. The key difference from regular parametric curves? You&#8217;re visualizing vectors in <strong>standard position<\/strong> (starting at the origin) rather than just points.<\/p>\n<p class=\"whitespace-normal break-words\">Think of it this way: as [latex]t[\/latex] increases, each vector [latex]{\\bf{r}}(t)[\/latex] points from the origin to a specific location in space. The <strong>terminal points<\/strong> (where these vectors end) trace out your curve. The curve&#8217;s shape depends on your function&#8217;s form:<\/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>Plane curve<\/strong>: [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}}[\/latex] creates a curve in the [latex]xy[\/latex]-plane<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Space curve<\/strong>: [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}} + h(t){\\bf{k}}[\/latex] creates a curve in three-dimensional space<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Graphing Strategy<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Make a table<\/strong>: Pick several [latex]t[\/latex]-values and calculate [latex]{\\bf{r}}(t)[\/latex] for each<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Draw vectors<\/strong>: Place each vector with its tail at the origin<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Mark endpoints<\/strong>: Note where each vector&#8217;s tip lands<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Connect smoothly<\/strong>: Draw a curve through these terminal points in order<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Show direction<\/strong>: Add arrows indicating how the curve progresses as [latex]t[\/latex] increases<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">Vector-valued functions and parametric equations are two ways of describing the same curve. Given [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}}[\/latex], you can write [latex]x = f(t)[\/latex] and [latex]y = g(t)[\/latex] to get the parametric form\u2014the graphs match exactly.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Reparameterization<\/strong> means different functions can trace the same curve. For example, [latex]{\\bf{r}}(t) = 4\\cos t{\\bf{i}} + 3\\sin t{\\bf{j}}[\/latex] and [latex]{\\bf{r}}(t) = 4\\cos t^3{\\bf{i}} + 3\\sin t^3{\\bf{j}}[\/latex] both produce ellipses\u2014the shape stays the same even though the parameter changes. Only the speed at which you trace the curve differs.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Special Case: Helices<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A <strong>helix<\/strong> is a three-dimensional spiral that looks like a spring. It has a circular or elliptical cross-section when viewed from above and climbs (or descends) as [latex]t[\/latex] increases. If you remove the [latex]{\\bf{k}[\/latex] component, you&#8217;re left with the curve&#8217;s projection onto the [latex]xy[\/latex]-plane.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Create a graph of the vector valued function [latex]{\\bf{r}}\\,(t)=(t^{2}-1)\\,{\\bf{i}}+(2t-3)\\,{\\bf{j}},\\ 0\\ \\leq\\ t\\ \\leq\\ 3.[\/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\"><\/div>\n<div>\n<figure id=\"attachment_5650\" aria-describedby=\"caption-attachment-5650\" style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5650\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2022\/04\/03163833\/Checkpoint-3.2.jpeg\" alt=\"This figure is a graph of the function r(t) = (t^2-1)i + (2t-3)j, for the values of t from 0 to 3. The curve begins in the 3rd quadrant at the ordered pair (-1,-3) and increases up through the 1st quadrant. It is increasing and has arrows on the curve representing orientation to the right.\" width=\"417\" height=\"347\" \/><figcaption id=\"caption-attachment-5650\" class=\"wp-caption-text\">Figure 4. Graph of [latex]{\\bf{r}}\\,(t)=(t^{2}-1)\\,{\\bf{i}}+(2t-3)\\,{\\bf{j}},\\ 0\\ \\leq\\ t\\ \\leq\\ 3[\/latex].<\/figcaption><\/figure>\n<\/div>\n<div class=\"exercise\"><\/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=7949574&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=nRScwQ-_TBU&amp;video_target=tpm-plugin-umi22qm5-nRScwQ-_TBU\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/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\/CP3.2_transcript.html\">transcript for \u201cCP 3.2\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<h2>Limits and Continuity of a Vector-Valued Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Finding limits and checking continuity for vector-valued functions works exactly like it does for regular functions\u2014you just handle each component separately.<\/p>\n<p class=\"whitespace-normal break-words\">Instead of wrestling with the formal definition involving [latex]|{\\bf{r}}(t) - {\\bf{L}}|[\/latex], use this practical method: <strong>take the limit of each component function individually<\/strong>.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>For plane curves<\/strong> [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}}[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\underset{t\\to a}{\\text{lim}}{\\bf{r}}(t)=\\bigg[\\underset{t\\to a}{\\text{lim}}f(t)\\bigg]{\\bf{i}}+\\bigg[\\underset{t\\to a}{\\text{lim}}g(t)\\bigg]{\\bf{j}}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>For space curves<\/strong> [latex]{\\bf{r}}(t) = f(t){\\bf{i}} + g(t){\\bf{j}} + h(t){\\bf{k}}[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\underset{t\\to a}{\\text{lim}}{\\bf{r}}(t)=\\bigg[\\underset{t\\to a}{\\text{lim}}f(t)\\bigg]{\\bf{i}}+\\bigg[\\underset{t\\to a}{\\text{lim}}g(t)\\bigg]{\\bf{j}}+\\bigg[\\underset{t\\to a}{\\text{lim}}h(t)\\bigg]{\\bf{k}}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This only works if <strong>all<\/strong> the component limits exist.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>How to Calculate Limits<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Break the vector function into its parts ([latex]{\\bf{i}}[\/latex], [latex]{\\bf{j}}[\/latex], and [latex]{\\bf{k}}[\/latex] components), find the limit of each scalar function using your usual techniques (substitution, algebraic manipulation, etc.), then reassemble the results into vector form. Each component follows the same limit rules you already know.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Checking Continuity<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A vector-valued function [latex]{\\bf{r}}(t)[\/latex] is continuous at [latex]t = a[\/latex] when these three familiar conditions hold:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]{\\bf{r}}(a)[\/latex] exists (the function is defined at that point)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\lim_{t\\to a}{\\bf{r}}(t)[\/latex] exists (the limit exists as you approach the point)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\lim_{t\\to a}{\\bf{r}}(t) = {\\bf{r}}(a)[\/latex] (the limit equals the function value)<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">These are the same three requirements you learned for regular functions. A vector-valued function is continuous at a point when its graph flows smoothly through that point without breaks, jumps, or gaps\u2014you could trace it without lifting your pencil.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Calculate [latex]\\displaystyle\\lim_{t{\\to}-2}{\\bf{r}}\\,(t)[\/latex] for the function [latex]{\\bf{r}}\\,(t)=\\sqrt{t^{2}-3t-1}\\,{\\bf{i}}+(4t+3)\\,{\\bf{j}}+\\sin{\\frac{(t+1)\\,\\pi}{2}}\\,{\\bf{k}}[\/latex].<\/p>\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\">\n<div>[latex]\\displaystyle\\lim_{t{\\to}-2}{\\bf{r}}\\,(t)=3\\,{\\bf{i}}-5\\,{\\bf{j}}-{\\bf{k}}[\/latex]<\/div>\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=7949584&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=UkThcL4VPfc&amp;video_target=tpm-plugin-x7y3bb4i-UkThcL4VPfc\" 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\/CP3.3_transcript.html\">transcript for \u201cCP 3.3\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-552","chapter","type-chapter","status-publish","hentry"],"part":424,"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\/552","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":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/552\/revisions"}],"predecessor-version":[{"id":2746,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/552\/revisions\/2746"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/424"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/552\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=552"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=552"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=552"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=552"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}