{"id":550,"date":"2025-10-09T15:20:52","date_gmt":"2025-10-09T15:20:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=550"},"modified":"2025-12-02T14:43:31","modified_gmt":"2025-12-02T14:43:31","slug":"introduction-to-vector-valued-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-vector-valued-functions-learn-it-1\/","title":{"raw":"Introduction to Vector-Valued Functions: Learn It 1","rendered":"Introduction to Vector-Valued Functions: Learn It 1"},"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<p class=\"whitespace-normal break-words\">Vector-valued functions combine ideas from single-variable calculus with our understanding of vectors in three dimensions. These functions allow us to describe curves in both 2D and 3D space mathematically, extending our calculus toolkit to handle motion and curves in space.<\/p>\r\n\r\n<h3>Definition of a Vector-Valued Function<\/h3>\r\n<p class=\"whitespace-normal break-words\">Instead of functions that output numbers (like [latex]f(t) = t^2[\/latex]), vector-valued functions output vectors. Think of them as functions that trace out paths or curves as the parameter [latex]t[\/latex] changes.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>vector-valued function<\/h3>\r\n<p class=\"whitespace-normal break-words\">A <strong><span id=\"term108\" data-type=\"term\">vector-valued function<\/span><\/strong> is a function of the form<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>2D<\/strong>:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{r}(t) = f(t)\\mathbf{i} + g(t)\\mathbf{j}[\/latex] or [latex]\\mathbf{r}(t) = \\langle f(t), g(t) \\rangle[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>3D<\/strong>:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{r}(t) = f(t)\\mathbf{i} + g(t)\\mathbf{j} + h(t)\\mathbf{k}[\/latex] or [latex]\\mathbf{r}(t) = \\langle f(t), g(t), h(t) \\rangle[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">where the\u00a0<strong><span id=\"term109\" data-type=\"term\">component functions<\/span> <\/strong> [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are real-valued functions of the parameter [latex]t[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">The parameter [latex]t[\/latex] often represents time, but it can represent any real variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each value of [latex]t[\/latex] produces a specific vector (point in space)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">As [latex]t[\/latex] varies, the vectors trace out a curve<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Domain and Range<\/h3>\r\n<p class=\"whitespace-normal break-words\">Understanding domain and range for vector-valued functions requires thinking about both the input parameter [latex]t[\/latex] and the vector outputs.<\/p>\r\n<p class=\"whitespace-normal break-words\">The <strong>domain<\/strong> of a vector-valued function is the set of all allowable values for the parameter [latex]t[\/latex]. Just like with regular functions, the domain consists of real numbers, but the restrictions come from all the component functions.<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>Determining the Domain<\/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\">Find the domain of each component function individually<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The domain of the vector-valued function is the <strong>intersection<\/strong> of all component domains<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Exclude any values where at least one component is undefined<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox connectIt\" aria-label=\"Connect It\">\r\n<p class=\"whitespace-normal break-words\"><strong>Common Domain Scenarios<\/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>No restrictions<\/strong>: When all components are polynomials, the domain is all real numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Bounded intervals<\/strong>: [latex]a \\leq t \\leq b[\/latex] when the problem specifies limits or when dealing with specific time intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Function-imposed restrictions<\/strong>: When components involve functions like [latex]\\tan t[\/latex], [latex]\\sec t[\/latex], [latex]\\ln t[\/latex], [latex]\\sqrt{t}[\/latex], or [latex]\\frac{1}{f(t)}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">If any component is undefined at [latex]t = c[\/latex], the entire vector function is undefined there.<\/p>\r\nThe <strong>range<\/strong> of a vector-valued function is fundamentally different from regular functions. Unlike regular functions that output numbers, vector-valued functions output 2D or 3D vectors.\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>2D functions<\/strong>: Range is a subset of [latex]\\mathbb{R}^2[\/latex] (set of 2D vectors)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>3D functions<\/strong>: Range is a subset of [latex]\\mathbb{R}^3[\/latex] (set of 3D vectors)<\/li>\r\n<\/ul>\r\nEach value [latex]t[\/latex] in the domain produces exactly one vector [latex]\\mathbf{r}(t)[\/latex]. As [latex]t[\/latex] varies over the entire domain, the collection of all resulting vectors forms the range.\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\u00a0The domain contains real numbers, but the range contains vectors. This fundamental difference makes vector-valued functions perfect for describing curves and motion in space.<\/section><section class=\"textbox example\" aria-label=\"Example\">For each of the following vector-valued functions, evaluate [latex]{\\bf{r}}(0), {\\bf{r}}(\\frac{\\pi}{2})[\/latex], and [latex]{\\bf{r}}(\\frac{2{\\pi}}{3})[\/latex].\u00a0Do any of these functions have domain restrictions?\r\n<ol type=\"a\">\r\n \t<li>[latex]{\\bf{r}}\\,(t)=4\\cos{t}\\,{\\bf{i}}+3\\sin{t}{\\bf{j}}[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{r}}\\,(t)=3\\tan{t}\\,{\\bf{i}}+4\\sec{t}{\\bf{j}}+5t{\\bf{k}}[\/latex]<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1167793361764\" class=\"exercise\">[reveal-answer q=\"fs-id1167794055154\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794055154\"]\r\n<ol type=\"a\">\r\n \t<li>To calculate each of the function values, substitute the appropriate value of <em data-effect=\"italics\">t<\/em> into the function:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\bf{r}}(0) &amp; =\\hfill &amp; 4\\cos{(0)}{\\bf{i}}+3\\sin{(0)}{\\bf{j}} \\hfill \\\\ \\hfill &amp; =\\hfill &amp; 4{\\bf{i}}+0{\\bf{j}}=4{\\bf{i}}\\hfill\\\\\\hfill{\\bf{r}}\\left(\\frac{\\pi}{2}\\right) &amp; =\\hfill &amp; 4\\cos{\\left(\\frac{\\pi}{2}\\right)}{\\bf{i}}+3\\sin{\\left(\\frac{\\pi}{2}\\right)}{\\bf{j}}\\hfill\\\\\\hfill&amp;=\\hfill&amp;0{\\bf{i}}+3{\\bf{j}}=3{\\bf{j}}\\hfill\\\\\\hfill{\\bf{r}}\\left(\\frac{2\\pi}{3}\\right)&amp; =\\hfill &amp; 4\\cos{\\left(\\frac{2\\pi}{3}\\right)}{\\bf{i}}+3\\sin{\\left(\\frac{2\\pi}{3}\\right)}{\\bf{j}}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; 4\\left(-\\frac{1}{2}\\right){\\bf{i}}+3\\left(\\frac{\\sqrt{3}}{2}\\right){\\bf{j}}=-2{\\bf{i}}+\\frac{3\\sqrt{3}}{2}{\\bf{j}}\\end{array}[\/latex].<\/div>\r\nTo determine whether this function has any domain restrictions, consider the component functions separately. The first component function is [latex]f(t)=4\\cos{t}[\/latex]\u00a0and the second component function is [latex]g(t)=3\\sin{t}[\/latex]\u00a0Neither of these functions has a domain restriction, so the domain of [latex]{\\bf{r}}(t)=4\\cos{t{\\bf{i}}}+3\\sin{t\\,{\\bf{j}}}[\/latex] is all real numbers.\r\n<div class=\"mceTemp\"><\/div><\/li>\r\n \t<li>To calculate each of the function values, substitute the appropriate value of [latex]t[\/latex] into the function:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\bf{r}}(0) &amp; =\\hfill &amp; 3\\tan{(0){\\bf{i}}}+4\\sec{(0){\\bf{j}}}+5\\,(0)\\,{\\bf{k}} \\hfill \\\\ \\hfill &amp; =\\hfill &amp; 0{\\bf{i}}+4{\\bf{j}}+0{\\bf{k}}=4{\\bf{j}}\\hfill\\\\\\hfill{\\bf{r}}\\left(\\frac{\\pi}{2}\\right) &amp; =\\hfill &amp; 3\\tan{\\left(\\frac{\\pi}{2}\\right){\\bf{i}}}+4\\sec{\\left(\\frac{\\pi}{2}\\right){\\bf{j}}}+5\\,\\left(\\frac{\\pi}{2}\\right)\\,{\\bf{k}},\\text{ which does not exist}\\hfill\\\\\\hfill{\\bf{r}}\\left(\\frac{2\\pi}{3}\\right) &amp; =\\hfill &amp; 3\\tan{\\left(\\frac{2\\pi}{2}\\right){\\bf{i}}}+4\\sec{\\left(\\frac{2\\pi}{2}\\right){\\bf{j}}}+5\\,\\left(\\frac{2\\pi}{2}\\right)\\,{\\bf{k}} \\hfill \\\\ \\hfill &amp; =\\hfill &amp; 3\\,(-\\sqrt{3})\\,{\\bf{i}}+4(-{2})\\,{\\bf{j}}+\\frac{10\\pi}{3}\\,{\\bf{k}} \\hfill \\\\ \\hfill &amp; =\\hfill &amp; -3\\sqrt{3}\\,{\\bf{i}}-8{\\bf{j}}+\\frac{10\\pi}{3}\\,{\\bf{k}}.\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\nTo determine whether this function has any domain restrictions, consider the component functions separately. The first component function is [latex]f\\,(t)=3\\tan{t}[\/latex],\u00a0the second component function is [latex]g\\,(t)=4\\sec{t}[\/latex], and the third component function is [latex]h\\,(t)=5t[\/latex].\u00a0The first two functions are not defined for odd multiples of [latex]\\frac{\\pi}{2}[\/latex], so the function is not defined for odd multiples of\u00a0[latex]\\frac{\\pi}{2}[\/latex]. Therefore, [latex]\\text{dom}({\\bf{r}}\\,(t))=\\bigg\\{t\\,\\bigg|\\,t\\neq\\frac{(2n+1){\\pi}}{2}\\bigg\\}[\/latex], where\u00a0[latex]n[\/latex] is any integer.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">For the vector-valued function [latex]{\\bf{r}}\\,(t)=(t^{2}-3t)\\,{\\bf{i}}+(4t+1)\\,{\\bf{j}}[\/latex], evaluate [latex]{\\bf{r}}\\,(0),\\ {\\bf{r}}\\,(1)[\/latex], and [latex]{\\bf{r}}\\,(-4)[\/latex]. Does this function have any domain restrictions?[reveal-answer q=\"fs-id1167793933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793933114\"]\r\n[latex]{\\bf{r}}\\,(0)={\\bf{j}},\\,{\\bf{r}}\\,(1)=-2\\,{\\bf{i}}+5\\,{\\bf{j}},\\,{\\bf{r}}\\,(-4)=28\\,{\\bf{i}}-15\\,{\\bf{j}}[\/latex]. The domain of\u00a0[latex]{\\bf{r}}\\,(t)=(t^{2}-3t)\\,{\\bf{i}}+(4t+1)\\,{\\bf{j}}[\/latex] is all real numbers.\r\n[\/hidden-answer]<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Geometric Interpretation<\/h3>\r\n<p class=\"whitespace-normal break-words\">Vector-valued functions have a powerful geometric interpretation: they describe <strong>parametric curves<\/strong> in space. As the parameter [latex]t[\/latex] varies over its domain, the tip of the vector [latex]\\mathbf{r}(t)[\/latex] traces out a curve.<\/p>\r\n<p class=\"whitespace-normal break-words\">For example, [latex]\\mathbf{r}(t) = 4\\cos t \\mathbf{i} + 3\\sin t \\mathbf{j}[\/latex] traces out an ellipse in the [latex]xy[\/latex]-plane as [latex]t[\/latex] goes from [latex]0[\/latex] to [latex]2\\pi[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">This connection between vector-valued functions and curves in space forms the foundation for studying motion, arc length, curvature, and many other important concepts in vector calculus.<\/p>","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<p class=\"whitespace-normal break-words\">Vector-valued functions combine ideas from single-variable calculus with our understanding of vectors in three dimensions. These functions allow us to describe curves in both 2D and 3D space mathematically, extending our calculus toolkit to handle motion and curves in space.<\/p>\n<h3>Definition of a Vector-Valued Function<\/h3>\n<p class=\"whitespace-normal break-words\">Instead of functions that output numbers (like [latex]f(t) = t^2[\/latex]), vector-valued functions output vectors. Think of them as functions that trace out paths or curves as the parameter [latex]t[\/latex] changes.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>vector-valued function<\/h3>\n<p class=\"whitespace-normal break-words\">A <strong><span id=\"term108\" data-type=\"term\">vector-valued function<\/span><\/strong> is a function of the form<\/p>\n<p class=\"whitespace-normal break-words\"><strong>2D<\/strong>:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{r}(t) = f(t)\\mathbf{i} + g(t)\\mathbf{j}[\/latex] or [latex]\\mathbf{r}(t) = \\langle f(t), g(t) \\rangle[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>3D<\/strong>:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{r}(t) = f(t)\\mathbf{i} + g(t)\\mathbf{j} + h(t)\\mathbf{k}[\/latex] or [latex]\\mathbf{r}(t) = \\langle f(t), g(t), h(t) \\rangle[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">where the\u00a0<strong><span id=\"term109\" data-type=\"term\">component functions<\/span> <\/strong> [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are real-valued functions of the parameter [latex]t[\/latex].<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<ul>\n<li class=\"whitespace-normal break-words\">The parameter [latex]t[\/latex] often represents time, but it can represent any real variable<\/li>\n<li class=\"whitespace-normal break-words\">Each value of [latex]t[\/latex] produces a specific vector (point in space)<\/li>\n<li class=\"whitespace-normal break-words\">As [latex]t[\/latex] varies, the vectors trace out a curve<\/li>\n<\/ul>\n<\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Domain and Range<\/h3>\n<p class=\"whitespace-normal break-words\">Understanding domain and range for vector-valued functions requires thinking about both the input parameter [latex]t[\/latex] and the vector outputs.<\/p>\n<p class=\"whitespace-normal break-words\">The <strong>domain<\/strong> of a vector-valued function is the set of all allowable values for the parameter [latex]t[\/latex]. Just like with regular functions, the domain consists of real numbers, but the restrictions come from all the component functions.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>Determining the Domain<\/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\">Find the domain of each component function individually<\/li>\n<li class=\"whitespace-normal break-words\">The domain of the vector-valued function is the <strong>intersection<\/strong> of all component domains<\/li>\n<li class=\"whitespace-normal break-words\">Exclude any values where at least one component is undefined<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\n<p class=\"whitespace-normal break-words\"><strong>Common Domain Scenarios<\/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>No restrictions<\/strong>: When all components are polynomials, the domain is all real numbers<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Bounded intervals<\/strong>: [latex]a \\leq t \\leq b[\/latex] when the problem specifies limits or when dealing with specific time intervals<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Function-imposed restrictions<\/strong>: When components involve functions like [latex]\\tan t[\/latex], [latex]\\sec t[\/latex], [latex]\\ln t[\/latex], [latex]\\sqrt{t}[\/latex], or [latex]\\frac{1}{f(t)}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p class=\"whitespace-normal break-words\">If any component is undefined at [latex]t = c[\/latex], the entire vector function is undefined there.<\/p>\n<p>The <strong>range<\/strong> of a vector-valued function is fundamentally different from regular functions. Unlike regular functions that output numbers, vector-valued functions output 2D or 3D vectors.<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>2D functions<\/strong>: Range is a subset of [latex]\\mathbb{R}^2[\/latex] (set of 2D vectors)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>3D functions<\/strong>: Range is a subset of [latex]\\mathbb{R}^3[\/latex] (set of 3D vectors)<\/li>\n<\/ul>\n<p>Each value [latex]t[\/latex] in the domain produces exactly one vector [latex]\\mathbf{r}(t)[\/latex]. As [latex]t[\/latex] varies over the entire domain, the collection of all resulting vectors forms the range.<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\u00a0The domain contains real numbers, but the range contains vectors. This fundamental difference makes vector-valued functions perfect for describing curves and motion in space.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">For each of the following vector-valued functions, evaluate [latex]{\\bf{r}}(0), {\\bf{r}}(\\frac{\\pi}{2})[\/latex], and [latex]{\\bf{r}}(\\frac{2{\\pi}}{3})[\/latex].\u00a0Do any of these functions have domain restrictions?<\/p>\n<ol type=\"a\">\n<li>[latex]{\\bf{r}}\\,(t)=4\\cos{t}\\,{\\bf{i}}+3\\sin{t}{\\bf{j}}[\/latex]<\/li>\n<li>[latex]{\\bf{r}}\\,(t)=3\\tan{t}\\,{\\bf{i}}+4\\sec{t}{\\bf{j}}+5t{\\bf{k}}[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1167793361764\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794055154\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794055154\" class=\"hidden-answer\" style=\"display: none\">\n<ol type=\"a\">\n<li>To calculate each of the function values, substitute the appropriate value of <em data-effect=\"italics\">t<\/em> into the function:\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\bf{r}}(0) & =\\hfill & 4\\cos{(0)}{\\bf{i}}+3\\sin{(0)}{\\bf{j}} \\hfill \\\\ \\hfill & =\\hfill & 4{\\bf{i}}+0{\\bf{j}}=4{\\bf{i}}\\hfill\\\\\\hfill{\\bf{r}}\\left(\\frac{\\pi}{2}\\right) & =\\hfill & 4\\cos{\\left(\\frac{\\pi}{2}\\right)}{\\bf{i}}+3\\sin{\\left(\\frac{\\pi}{2}\\right)}{\\bf{j}}\\hfill\\\\\\hfill&=\\hfill&0{\\bf{i}}+3{\\bf{j}}=3{\\bf{j}}\\hfill\\\\\\hfill{\\bf{r}}\\left(\\frac{2\\pi}{3}\\right)& =\\hfill & 4\\cos{\\left(\\frac{2\\pi}{3}\\right)}{\\bf{i}}+3\\sin{\\left(\\frac{2\\pi}{3}\\right)}{\\bf{j}}\\hfill \\\\ \\hfill & =\\hfill & 4\\left(-\\frac{1}{2}\\right){\\bf{i}}+3\\left(\\frac{\\sqrt{3}}{2}\\right){\\bf{j}}=-2{\\bf{i}}+\\frac{3\\sqrt{3}}{2}{\\bf{j}}\\end{array}[\/latex].<\/div>\n<p>To determine whether this function has any domain restrictions, consider the component functions separately. The first component function is [latex]f(t)=4\\cos{t}[\/latex]\u00a0and the second component function is [latex]g(t)=3\\sin{t}[\/latex]\u00a0Neither of these functions has a domain restriction, so the domain of [latex]{\\bf{r}}(t)=4\\cos{t{\\bf{i}}}+3\\sin{t\\,{\\bf{j}}}[\/latex] is all real numbers.<\/p>\n<div class=\"mceTemp\"><\/div>\n<\/li>\n<li>To calculate each of the function values, substitute the appropriate value of [latex]t[\/latex] into the function:\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\bf{r}}(0) & =\\hfill & 3\\tan{(0){\\bf{i}}}+4\\sec{(0){\\bf{j}}}+5\\,(0)\\,{\\bf{k}} \\hfill \\\\ \\hfill & =\\hfill & 0{\\bf{i}}+4{\\bf{j}}+0{\\bf{k}}=4{\\bf{j}}\\hfill\\\\\\hfill{\\bf{r}}\\left(\\frac{\\pi}{2}\\right) & =\\hfill & 3\\tan{\\left(\\frac{\\pi}{2}\\right){\\bf{i}}}+4\\sec{\\left(\\frac{\\pi}{2}\\right){\\bf{j}}}+5\\,\\left(\\frac{\\pi}{2}\\right)\\,{\\bf{k}},\\text{ which does not exist}\\hfill\\\\\\hfill{\\bf{r}}\\left(\\frac{2\\pi}{3}\\right) & =\\hfill & 3\\tan{\\left(\\frac{2\\pi}{2}\\right){\\bf{i}}}+4\\sec{\\left(\\frac{2\\pi}{2}\\right){\\bf{j}}}+5\\,\\left(\\frac{2\\pi}{2}\\right)\\,{\\bf{k}} \\hfill \\\\ \\hfill & =\\hfill & 3\\,(-\\sqrt{3})\\,{\\bf{i}}+4(-{2})\\,{\\bf{j}}+\\frac{10\\pi}{3}\\,{\\bf{k}} \\hfill \\\\ \\hfill & =\\hfill & -3\\sqrt{3}\\,{\\bf{i}}-8{\\bf{j}}+\\frac{10\\pi}{3}\\,{\\bf{k}}.\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\n<p>To determine whether this function has any domain restrictions, consider the component functions separately. The first component function is [latex]f\\,(t)=3\\tan{t}[\/latex],\u00a0the second component function is [latex]g\\,(t)=4\\sec{t}[\/latex], and the third component function is [latex]h\\,(t)=5t[\/latex].\u00a0The first two functions are not defined for odd multiples of [latex]\\frac{\\pi}{2}[\/latex], so the function is not defined for odd multiples of\u00a0[latex]\\frac{\\pi}{2}[\/latex]. Therefore, [latex]\\text{dom}({\\bf{r}}\\,(t))=\\bigg\\{t\\,\\bigg|\\,t\\neq\\frac{(2n+1){\\pi}}{2}\\bigg\\}[\/latex], where\u00a0[latex]n[\/latex] is any integer.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">For the vector-valued function [latex]{\\bf{r}}\\,(t)=(t^{2}-3t)\\,{\\bf{i}}+(4t+1)\\,{\\bf{j}}[\/latex], evaluate [latex]{\\bf{r}}\\,(0),\\ {\\bf{r}}\\,(1)[\/latex], and [latex]{\\bf{r}}\\,(-4)[\/latex]. Does this function have any domain restrictions?<\/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[latex]{\\bf{r}}\\,(0)={\\bf{j}},\\,{\\bf{r}}\\,(1)=-2\\,{\\bf{i}}+5\\,{\\bf{j}},\\,{\\bf{r}}\\,(-4)=28\\,{\\bf{i}}-15\\,{\\bf{j}}[\/latex]. The domain of\u00a0[latex]{\\bf{r}}\\,(t)=(t^{2}-3t)\\,{\\bf{i}}+(4t+1)\\,{\\bf{j}}[\/latex] is all real numbers.\n<\/div>\n<\/div>\n<\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Geometric Interpretation<\/h3>\n<p class=\"whitespace-normal break-words\">Vector-valued functions have a powerful geometric interpretation: they describe <strong>parametric curves<\/strong> in space. As the parameter [latex]t[\/latex] varies over its domain, the tip of the vector [latex]\\mathbf{r}(t)[\/latex] traces out a curve.<\/p>\n<p class=\"whitespace-normal break-words\">For example, [latex]\\mathbf{r}(t) = 4\\cos t \\mathbf{i} + 3\\sin t \\mathbf{j}[\/latex] traces out an ellipse in the [latex]xy[\/latex]-plane as [latex]t[\/latex] goes from [latex]0[\/latex] to [latex]2\\pi[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">This connection between vector-valued functions and curves in space forms the foundation for studying motion, arc length, curvature, and many other important concepts in vector calculus.<\/p>\n","protected":false},"author":15,"menu_order":6,"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-550","chapter","type-chapter","status-publish","hentry"],"part":424,"module-header":"learn_it","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\/550","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\/550\/revisions"}],"predecessor-version":[{"id":2744,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/550\/revisions\/2744"}],"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\/550\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=550"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=550"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=550"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=550"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}