{"id":3849,"date":"2022-04-04T15:59:46","date_gmt":"2022-04-04T15:59:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=3849"},"modified":"2022-10-29T00:31:34","modified_gmt":"2022-10-29T00:31:34","slug":"vector-valued-functions-and-space-curves","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/vector-valued-functions-and-space-curves\/","title":{"raw":"Vector-Valued Functions and Space Curves","rendered":"Vector-Valued Functions and Space Curves"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Write the general equation of a vector-valued function in component form and unit-vector form.<\/li>\r\n \t<li>Recognize parametric equations for a space curve.<\/li>\r\n \t<li>Describe the shape of a helix and write its equation.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Definition of a Vector-Valued Function<\/h2>\r\nOur first step in studying the calculus of vector-valued functions is to define what exactly a vector-valued function is. We can then look at graphs of vector-valued functions and see how they define curves in both two and three dimensions.\r\n<div id=\"fs-id1167793372222\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\r\nA <strong><span id=\"term108\" data-type=\"term\">vector-valued function<\/span><\/strong> is a function of the form\r\n<div style=\"text-align: center;\">[latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}} + g\\,(t)\\,{\\bf{j}}\\;\\text{ or }\\;{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}} + g\\,(t)\\,{\\bf{j}} + h\\,(t)\\,{\\bf{k}}[\/latex],<\/div>\r\n&nbsp;\r\n\r\nwhere 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]<em data-effect=\"italics\">.<\/em> Vector-valued functions are also written in the form\r\n<div style=\"text-align: center;\">[latex]{\\bf{r}}\\,(t)={\\langle}f\\,(t),\\ g\\,(t){\\rangle}\\;\\text{ or }\\;{\\bf{r}}\\,(t)={\\langle}f\\,(t),\\ g\\,(t),\\ h\\,(t){\\rangle}[\/latex].<\/div>\r\n&nbsp;\r\n\r\nIn both cases, the first form of the function defines a two-dimensional vector-valued function; the second form describes a three-dimensional vector-valued function.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe parameter [latex]t[\/latex]\u00a0can lie between two real numbers: [latex]a \\leq t \\leq b[\/latex]. Another possibility is that the value of [latex]t[\/latex] might take on all real numbers. Last, the component functions themselves may have domain restrictions that enforce restrictions on the value of [latex]t[\/latex]<em data-effect=\"italics\">.<\/em> We often use [latex]t[\/latex] as a parameter because [latex]t[\/latex] can represent time.\r\n<div id=\"fs-id1167793900960\" class=\"textbook exercises\">\r\n<h3>Example:\u00a0Evaluating Vector-Valued Functions and Determining Domains<\/h3>\r\nFor 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<\/div>\r\n<div id=\"fs-id1167793957091\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFor 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?\r\n\r\n[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]\r\n\r\n<\/div>\r\nThe above example illustrates an important concept. The domain of a vector-valued function consists of real numbers. The domain can be all real numbers or a subset of the real numbers. The range of a vector-valued function consists of vectors. Each real number in the domain of a vector-valued function is mapped to either a two- or a three-dimensional vector.\r\n<h2>Graphing Vector-Valued Functions<\/h2>\r\n<p id=\"fs-id1169737711899\" class=\" \">Recall that a plane vector consists of two quantities: direction and magnitude. Given any point in the plane (the <span id=\"term110\" class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">initial point<\/em><\/span>), if we move in a specific direction for a specific distance, we arrive at a second point. This represents the <span id=\"term111\" class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">terminal point<\/em><\/span> of the vector. We calculate the components of the vector by subtracting the coordinates of the initial point from the coordinates of the terminal point.<\/p>\r\n<p id=\"fs-id1169738221453\" class=\" \">A vector is considered to be in <span id=\"term112\" class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">standard position<\/em><\/span> if the initial point is located at the origin. When graphing a vector-valued function, we typically graph the vectors in the domain of the function in standard position, because doing so guarantees the uniqueness of the graph. This convention applies to the graphs of three-dimensional vector-valued functions as well. The graph of a vector-valued function of the form [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}[\/latex]\u00a0consists of the set of all [latex](t,\\ {\\bf{r}}\\,(t))[\/latex], and the path it traces is called a <strong><span id=\"term113\" data-type=\"term\">plane curve<\/span><\/strong>. The graph of a vector-valued function of the form [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}+h\\,(t)\\,{\\bf{k}}[\/latex] consists of the set of all [latex](t,\\ {\\bf{r}}\\,(t))[\/latex]\u00a0and the path it traces is called a <strong><span id=\"term114\" data-type=\"term\">space curve<\/span><\/strong>. Any representation of a plane curve or space curve using a vector-valued function is called a <strong><span id=\"term115\" data-type=\"term\">vector parameterization<\/span><\/strong> of the curve.<\/p>\r\n\r\n<div id=\"fs-id1167793900968\" class=\"textbook exercises\">\r\n<h3>Example: Combining Vectors<\/h3>\r\nCreate a graph of each of the following vector-valued functions:\r\n<ol>\r\n \t<li>[latex]{\\bf{r}}\\,(t)=4\\cos{t{\\bf{i}}}+3\\sin{t{\\bf{j}}},\\ 0\\,\\leq\\,t\\,\\leq\\,2\\pi[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{r}}\\,(t)=4\\cos{t^{3}{\\bf{i}}}+3\\sin{t^{3}\\,{\\bf{j}}},\\ 0\\,\\leq\\,t\\,\\leq\\,2\\pi[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{r}}\\,(t)=\\cos{t\\,{\\bf{i}}}+\\sin{t\\,{\\bf{j}}}+t\\,{\\bf{k}},\\ 0\\,\\leq\\,t\\,\\leq\\,4\\pi[\/latex]<\/li>\r\n<\/ol>\r\n<div class=\"mceTemp\"><\/div>\r\n<div><\/div>\r\n<div id=\"fs-id1167793361764\" class=\"exercise\">[reveal-answer q=\"fs-id1167794055165\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794055165\"]\r\n<ol>\r\n \t<li>As with any graph, we start with a table of values. We then graph each of the vectors in the second column of the table in standard position and connect the terminal points of each vector to form a curve. This curve turns out to be an ellipse centered at the origin.\r\n<div>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 112px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">[latex]t[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]t[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]\\pi[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]-4\\,{\\bf{i}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 25%; height: 28px;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 28px;\">[latex]2\\sqrt{2}\\,{\\bf{i}}+\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 28px;\">[latex]\\frac{5\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 28px;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}-\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]3\\,{\\bf{j}}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]\\frac{3\\pi}{2}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]-3\\,{\\bf{j}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 25%; height: 28px;\">[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 28px;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}+\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 28px;\">[latex]\\frac{7\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 28px;\">[latex]2\\sqrt{2}\\,{\\bf{i}}-\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px;\">[latex]2\\pi[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px;\"><\/td>\r\n<td style=\"width: 25%; height: 14px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div>\r\n\r\n[caption id=\"attachment_901\" align=\"aligncenter\" width=\"417\"]<img class=\"wp-image-901 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21155059\/3-1-1.jpeg\" alt=\"This figure is a graph of an ellipse centered at the origin. The graph is the vector-valued function r(t)=4cost i + 3sint j. The ellipse has arrows on the curve representing counter-clockwise orientation. There are also line segments inside of the ellipse to the curve at different increments of t. The increments are t=0, t=pi\/4, t=pi\/2, t=3pi\/4.\" width=\"417\" height=\"422\" \/> Figure 1.\u00a0The graph of the first vector-valued function is an ellipse.[\/caption]\r\n\r\n<\/div>\r\n<div><\/div><\/li>\r\n \t<li>The table of values for [latex]{\\bf{r}}\\,(t)=4\\cos{t\\,{\\bf{i}}}+3\\sin{t\\,{\\bf{j}}},\\ 0\\ \\leq\\ t\\ \\leq\\ 2\\pi[\/latex] is as follows:\r\n<div>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]-4\\,{\\bf{i}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{8}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]2\\sqrt{2}\\,{\\bf{i}}+\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]\\frac{5\\pi}{8}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}-\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]3\\,{\\bf{j}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]-3\\,{\\bf{j}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]\\frac{3\\pi}{8}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}+\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]\\frac{7\\pi}{8}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]2\\sqrt{2}\\,{\\bf{i}}-\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]\\pi[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\"><\/td>\r\n<td style=\"width: 25%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div>The graph of this curve is also an ellipse centered at the origin.<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"CNX_Calc_Figure_13_01_002\" class=\"os-figure\">\r\n\r\n[caption id=\"attachment_903\" align=\"aligncenter\" width=\"425\"]<img class=\"size-full wp-image-903\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21155329\/3-1-2.jpeg\" alt=\"This figure is a graph of an ellipse centered at the origin. The graph is the vector-valued function r(t)=4cost^3 i + 3sint^3 j.\" width=\"425\" height=\"422\" \/> Figure 2. The graph of the second vector-valued function is also an ellipse.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div><\/div><\/li>\r\n \t<li>We go through the same procedure for a three-dimensional vector function.\r\n<div>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]\\pi[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]-4\\,{\\bf{j}}+\\pi\\,{\\bf{k}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]2\\sqrt{2}\\,{\\bf{i}}+2\\sqrt{2}\\,{\\bf{j}}+\\frac{\\pi}{4}\\,{\\bf{k}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]\\frac{5\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}-2\\sqrt{2}\\,{\\bf{j}}+\\frac{5\\pi}{4}\\,{\\bf{k}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{j}}+\\frac{\\pi}{2}{\\bf{k}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]\\frac{3\\pi}{2}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]-4\\,{\\bf{j}}+\\frac{3\\pi}{2}\\,{\\bf{k}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}+2\\sqrt{2}\\,{\\bf{j}}+\\frac{3\\pi}{4}\\,{\\bf{k}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]\\frac{7\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]2\\sqrt{2}\\,{\\bf{i}}-2\\sqrt{2}\\,{\\bf{j}}+\\frac{7\\pi}{4}\\,{\\bf{k}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]2\\pi[\/latex]<\/td>\r\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{i}}+2\\pi\\,{\\bf{k}}[\/latex]<\/td>\r\n<td style=\"width: 25%;\"><\/td>\r\n<td style=\"width: 25%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div>The values then repeat themselves, except for the fact that the coefficient of k is always increasing (Figure 3.4). This curve is called a <strong>helix<\/strong>. Notice that if the<strong> k<\/strong> component is eliminated, then the function becomes [latex]{\\bf{r}}\\,(t)=\\cos{t\\,{\\bf{i}}}+\\sin{t\\,{\\bf{j}}}[\/latex], which is a unit circle centered at the origin.<\/div>\r\n<div><\/div>\r\n<div>\r\n\r\n[caption id=\"attachment_904\" align=\"aligncenter\" width=\"393\"]<img class=\"size-full wp-image-904\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21155435\/3-1-3.jpeg\" alt=\"This figure is the graph of a helix in the 3 dimensional coordinate system. The curve represents the function r(t) = cost i + sint j + tk. The curve spirals in a circular path around the vertical z-axis and has the look of a spring. The arrows on the curve represent orientation.\" width=\"393\" height=\"486\" \/> Figure 3.\u00a0The graph of the third vector-valued function is a helix.[\/caption]\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nYou may notice that the graphs in parts 1 and 2 are identical. This happens because the function describing curve 2 is a so-called <strong><span id=\"term117\" data-type=\"term\">reparameterization<\/span><\/strong> of the function describing curve 1. In fact, any curve has an infinite number of reparameterizations; for example, we can replace [latex]t[\/latex] with [latex]2t[\/latex]\u00a0in any of the three previous curves without changing the shape of the curve. The interval over which <em data-effect=\"italics\">t<\/em> is defined may change, but that is all. We return to this idea later in this chapter when we study arc-length parameterization.\r\n\r\nAs mentioned, the name of the shape of the curve of the graph in part 3 of the previous example\u00a0is a <strong><span id=\"term118\" data-type=\"term\">helix<\/span><\/strong> (Figure 3). The curve resembles a spring, with a circular cross-section looking down along the [latex]z[\/latex]-axis. It is possible for a helix to be elliptical in cross-section as well. For example, the vector-valued function [latex]{\\bf{r}}\\,(t)=4\\cos{t\\,{\\bf{i}}}+3\\sin{t\\,{\\bf{j}}}+t\\,{\\bf{k}}[\/latex]\u00a0describes an elliptical helix. The projection of this helix into the [latex]x,y\\text{-plane}[\/latex] is an ellipse.\u00a0Last, the arrows in the graph of this helix indicate the orientation of the curve as [latex]t[\/latex]\u00a0progresses from [latex]0[\/latex] to [latex]4\\pi[\/latex].\r\n<div id=\"fs-id1167793957097\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nCreate 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<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<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>At this point, you may notice a similarity between vector-valued functions and parameterized curves. Indeed, given a vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}[\/latex], we can define [latex]x=f\\,(t)[\/latex] and [latex]y=g\\,(t)[\/latex].\u00a0If a restriction exists on the values of [latex]t[\/latex] (for example, [latex]t[\/latex]\u00a0is restricted to the interval [latex][a,\\ b][\/latex] for some constants [latex]a\\ \\le\\ b[\/latex]). then this restriction is enforced on the parameter\u00a0The graph of the parameterized function would then agree with the graph of the vector-valued function, except that the vector-valued graph would represent vectors rather than points. Since we can parameterize a curve defined by a function [latex]y=f\\,(x)[\/latex], it is also possible to represent an arbitrary plane curve by a vector-valued function.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write the general equation of a vector-valued function in component form and unit-vector form.<\/li>\n<li>Recognize parametric equations for a space curve.<\/li>\n<li>Describe the shape of a helix and write its equation.<\/li>\n<\/ul>\n<\/div>\n<h2>Definition of a Vector-Valued Function<\/h2>\n<p>Our first step in studying the calculus of vector-valued functions is to define what exactly a vector-valued function is. We can then look at graphs of vector-valued functions and see how they define curves in both two and three dimensions.<\/p>\n<div id=\"fs-id1167793372222\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p>A <strong><span id=\"term108\" data-type=\"term\">vector-valued function<\/span><\/strong> is a function of the form<\/p>\n<div style=\"text-align: center;\">[latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}} + g\\,(t)\\,{\\bf{j}}\\;\\text{ or }\\;{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}} + g\\,(t)\\,{\\bf{j}} + h\\,(t)\\,{\\bf{k}}[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p>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]<em data-effect=\"italics\">.<\/em> Vector-valued functions are also written in the form<\/p>\n<div style=\"text-align: center;\">[latex]{\\bf{r}}\\,(t)={\\langle}f\\,(t),\\ g\\,(t){\\rangle}\\;\\text{ or }\\;{\\bf{r}}\\,(t)={\\langle}f\\,(t),\\ g\\,(t),\\ h\\,(t){\\rangle}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>In both cases, the first form of the function defines a two-dimensional vector-valued function; the second form describes a three-dimensional vector-valued function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The parameter [latex]t[\/latex]\u00a0can lie between two real numbers: [latex]a \\leq t \\leq b[\/latex]. Another possibility is that the value of [latex]t[\/latex] might take on all real numbers. Last, the component functions themselves may have domain restrictions that enforce restrictions on the value of [latex]t[\/latex]<em data-effect=\"italics\">.<\/em> We often use [latex]t[\/latex] as a parameter because [latex]t[\/latex] can represent time.<\/p>\n<div id=\"fs-id1167793900960\" class=\"textbook exercises\">\n<h3>Example:\u00a0Evaluating Vector-Valued Functions and Determining Domains<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794055154\">Show Solution<\/span><\/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<\/div>\n<div id=\"fs-id1167793957091\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793933114\">Show Solution<\/span><\/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<\/div>\n<p>The above example illustrates an important concept. The domain of a vector-valued function consists of real numbers. The domain can be all real numbers or a subset of the real numbers. The range of a vector-valued function consists of vectors. Each real number in the domain of a vector-valued function is mapped to either a two- or a three-dimensional vector.<\/p>\n<h2>Graphing Vector-Valued Functions<\/h2>\n<p id=\"fs-id1169737711899\" class=\"\">Recall that a plane vector consists of two quantities: direction and magnitude. Given any point in the plane (the <span id=\"term110\" class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">initial point<\/em><\/span>), if we move in a specific direction for a specific distance, we arrive at a second point. This represents the <span id=\"term111\" class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">terminal point<\/em><\/span> of the vector. We calculate the components of the vector by subtracting the coordinates of the initial point from the coordinates of the terminal point.<\/p>\n<p id=\"fs-id1169738221453\" class=\"\">A vector is considered to be in <span id=\"term112\" class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">standard position<\/em><\/span> if the initial point is located at the origin. When graphing a vector-valued function, we typically graph the vectors in the domain of the function in standard position, because doing so guarantees the uniqueness of the graph. This convention applies to the graphs of three-dimensional vector-valued functions as well. The graph of a vector-valued function of the form [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}[\/latex]\u00a0consists of the set of all [latex](t,\\ {\\bf{r}}\\,(t))[\/latex], and the path it traces is called a <strong><span id=\"term113\" data-type=\"term\">plane curve<\/span><\/strong>. The graph of a vector-valued function of the form [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}+h\\,(t)\\,{\\bf{k}}[\/latex] consists of the set of all [latex](t,\\ {\\bf{r}}\\,(t))[\/latex]\u00a0and the path it traces is called a <strong><span id=\"term114\" data-type=\"term\">space curve<\/span><\/strong>. Any representation of a plane curve or space curve using a vector-valued function is called a <strong><span id=\"term115\" data-type=\"term\">vector parameterization<\/span><\/strong> of the curve.<\/p>\n<div id=\"fs-id1167793900968\" class=\"textbook exercises\">\n<h3>Example: Combining Vectors<\/h3>\n<p>Create a graph of each of the following vector-valued functions:<\/p>\n<ol>\n<li>[latex]{\\bf{r}}\\,(t)=4\\cos{t{\\bf{i}}}+3\\sin{t{\\bf{j}}},\\ 0\\,\\leq\\,t\\,\\leq\\,2\\pi[\/latex]<\/li>\n<li>[latex]{\\bf{r}}\\,(t)=4\\cos{t^{3}{\\bf{i}}}+3\\sin{t^{3}\\,{\\bf{j}}},\\ 0\\,\\leq\\,t\\,\\leq\\,2\\pi[\/latex]<\/li>\n<li>[latex]{\\bf{r}}\\,(t)=\\cos{t\\,{\\bf{i}}}+\\sin{t\\,{\\bf{j}}}+t\\,{\\bf{k}},\\ 0\\,\\leq\\,t\\,\\leq\\,4\\pi[\/latex]<\/li>\n<\/ol>\n<div class=\"mceTemp\"><\/div>\n<div><\/div>\n<div id=\"fs-id1167793361764\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794055165\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794055165\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>As with any graph, we start with a table of values. We then graph each of the vectors in the second column of the table in standard position and connect the terminal points of each vector to form a curve. This curve turns out to be an ellipse centered at the origin.\n<div>\n<table style=\"border-collapse: collapse; width: 100%; height: 112px;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]\\pi[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]-4\\,{\\bf{i}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"width: 25%; height: 28px;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%; height: 28px;\">[latex]2\\sqrt{2}\\,{\\bf{i}}+\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\n<td style=\"width: 25%; height: 28px;\">[latex]\\frac{5\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%; height: 28px;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}-\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]3\\,{\\bf{j}}[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]\\frac{3\\pi}{2}[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]-3\\,{\\bf{j}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"width: 25%; height: 28px;\">[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%; height: 28px;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}+\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\n<td style=\"width: 25%; height: 28px;\">[latex]\\frac{7\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%; height: 28px;\">[latex]2\\sqrt{2}\\,{\\bf{i}}-\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px;\">[latex]2\\pi[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px;\"><\/td>\n<td style=\"width: 25%; height: 14px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div>\n<div id=\"attachment_901\" style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-901\" class=\"wp-image-901 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21155059\/3-1-1.jpeg\" alt=\"This figure is a graph of an ellipse centered at the origin. The graph is the vector-valued function r(t)=4cost i + 3sint j. The ellipse has arrows on the curve representing counter-clockwise orientation. There are also line segments inside of the ellipse to the curve at different increments of t. The increments are t=0, t=pi\/4, t=pi\/2, t=3pi\/4.\" width=\"417\" height=\"422\" \/><\/p>\n<p id=\"caption-attachment-901\" class=\"wp-caption-text\">Figure 1.\u00a0The graph of the first vector-valued function is an ellipse.<\/p>\n<\/div>\n<\/div>\n<div><\/div>\n<\/li>\n<li>The table of values for [latex]{\\bf{r}}\\,(t)=4\\cos{t\\,{\\bf{i}}}+3\\sin{t\\,{\\bf{j}}},\\ 0\\ \\leq\\ t\\ \\leq\\ 2\\pi[\/latex] is as follows:\n<div>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-4\\,{\\bf{i}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{8}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]2\\sqrt{2}\\,{\\bf{i}}+\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]\\frac{5\\pi}{8}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}-\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]3\\,{\\bf{j}}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-3\\,{\\bf{j}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]\\frac{3\\pi}{8}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}+\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]\\frac{7\\pi}{8}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]2\\sqrt{2}\\,{\\bf{i}}-\\frac{3\\sqrt{2}}{2}\\,{\\bf{j}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]\\pi[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\n<td style=\"width: 25%;\"><\/td>\n<td style=\"width: 25%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div>The graph of this curve is also an ellipse centered at the origin.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"CNX_Calc_Figure_13_01_002\" class=\"os-figure\">\n<div id=\"attachment_903\" style=\"width: 435px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-903\" class=\"size-full wp-image-903\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21155329\/3-1-2.jpeg\" alt=\"This figure is a graph of an ellipse centered at the origin. The graph is the vector-valued function r(t)=4cost^3 i + 3sint^3 j.\" width=\"425\" height=\"422\" \/><\/p>\n<p id=\"caption-attachment-903\" class=\"wp-caption-text\">Figure 2. The graph of the second vector-valued function is also an ellipse.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<\/li>\n<li>We go through the same procedure for a three-dimensional vector function.\n<div>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]{\\bf{r}}\\,(t)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{i}}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]\\pi[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-4\\,{\\bf{j}}+\\pi\\,{\\bf{k}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]2\\sqrt{2}\\,{\\bf{i}}+2\\sqrt{2}\\,{\\bf{j}}+\\frac{\\pi}{4}\\,{\\bf{k}}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]\\frac{5\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}-2\\sqrt{2}\\,{\\bf{j}}+\\frac{5\\pi}{4}\\,{\\bf{k}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{j}}+\\frac{\\pi}{2}{\\bf{k}}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]\\frac{3\\pi}{2}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-4\\,{\\bf{j}}+\\frac{3\\pi}{2}\\,{\\bf{k}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]-2\\sqrt{2}\\,{\\bf{i}}+2\\sqrt{2}\\,{\\bf{j}}+\\frac{3\\pi}{4}\\,{\\bf{k}}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]\\frac{7\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]2\\sqrt{2}\\,{\\bf{i}}-2\\sqrt{2}\\,{\\bf{j}}+\\frac{7\\pi}{4}\\,{\\bf{k}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]2\\pi[\/latex]<\/td>\n<td style=\"width: 25%;\">[latex]4\\,{\\bf{i}}+2\\pi\\,{\\bf{k}}[\/latex]<\/td>\n<td style=\"width: 25%;\"><\/td>\n<td style=\"width: 25%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div>The values then repeat themselves, except for the fact that the coefficient of k is always increasing (Figure 3.4). This curve is called a <strong>helix<\/strong>. Notice that if the<strong> k<\/strong> component is eliminated, then the function becomes [latex]{\\bf{r}}\\,(t)=\\cos{t\\,{\\bf{i}}}+\\sin{t\\,{\\bf{j}}}[\/latex], which is a unit circle centered at the origin.<\/div>\n<div><\/div>\n<div>\n<div id=\"attachment_904\" style=\"width: 403px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-904\" class=\"size-full wp-image-904\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/21155435\/3-1-3.jpeg\" alt=\"This figure is the graph of a helix in the 3 dimensional coordinate system. The curve represents the function r(t) = cost i + sint j + tk. The curve spirals in a circular path around the vertical z-axis and has the look of a spring. The arrows on the curve represent orientation.\" width=\"393\" height=\"486\" \/><\/p>\n<p id=\"caption-attachment-904\" class=\"wp-caption-text\">Figure 3.\u00a0The graph of the third vector-valued function is a helix.<\/p>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>You may notice that the graphs in parts 1 and 2 are identical. This happens because the function describing curve 2 is a so-called <strong><span id=\"term117\" data-type=\"term\">reparameterization<\/span><\/strong> of the function describing curve 1. In fact, any curve has an infinite number of reparameterizations; for example, we can replace [latex]t[\/latex] with [latex]2t[\/latex]\u00a0in any of the three previous curves without changing the shape of the curve. The interval over which <em data-effect=\"italics\">t<\/em> is defined may change, but that is all. We return to this idea later in this chapter when we study arc-length parameterization.<\/p>\n<p>As mentioned, the name of the shape of the curve of the graph in part 3 of the previous example\u00a0is a <strong><span id=\"term118\" data-type=\"term\">helix<\/span><\/strong> (Figure 3). The curve resembles a spring, with a circular cross-section looking down along the [latex]z[\/latex]-axis. It is possible for a helix to be elliptical in cross-section as well. For example, the vector-valued function [latex]{\\bf{r}}\\,(t)=4\\cos{t\\,{\\bf{i}}}+3\\sin{t\\,{\\bf{j}}}+t\\,{\\bf{k}}[\/latex]\u00a0describes an elliptical helix. The projection of this helix into the [latex]x,y\\text{-plane}[\/latex] is an ellipse.\u00a0Last, the arrows in the graph of this helix indicate the orientation of the curve as [latex]t[\/latex]\u00a0progresses from [latex]0[\/latex] to [latex]4\\pi[\/latex].<\/p>\n<div id=\"fs-id1167793957097\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793933124\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793933124\" class=\"hidden-answer\" style=\"display: none\"><\/div>\n<div>\n<div id=\"attachment_5650\" style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5650\" 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\" \/><\/p>\n<p 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].<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>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=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<p>At this point, you may notice a similarity between vector-valued functions and parameterized curves. Indeed, given a vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}[\/latex], we can define [latex]x=f\\,(t)[\/latex] and [latex]y=g\\,(t)[\/latex].\u00a0If a restriction exists on the values of [latex]t[\/latex] (for example, [latex]t[\/latex]\u00a0is restricted to the interval [latex][a,\\ b][\/latex] for some constants [latex]a\\ \\le\\ b[\/latex]). then this restriction is enforced on the parameter\u00a0The graph of the parameterized function would then agree with the graph of the vector-valued function, except that the vector-valued graph would represent vectors rather than points. Since we can parameterize a curve defined by a function [latex]y=f\\,(x)[\/latex], it is also possible to represent an arbitrary plane curve by a vector-valued function.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3849\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 3.2. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"CP 3.2\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3849","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/3849","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/3849\/revisions"}],"predecessor-version":[{"id":6444,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/3849\/revisions\/6444"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/3849\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=3849"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=3849"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=3849"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=3849"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}