{"id":948,"date":"2025-10-14T16:31:32","date_gmt":"2025-10-14T16:31:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=948"},"modified":"2025-10-22T16:13:01","modified_gmt":"2025-10-22T16:13:01","slug":"introduction-to-vector-valued-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-vector-valued-functions-learn-it-2\/","title":{"raw":"Introduction to Vector-Valued Functions: Learn It 2","rendered":"Introduction to Vector-Valued Functions: Learn It 2"},"content":{"raw":"<h2>Graphing Vector-Valued Functions<\/h2>\r\nA plane vector has two components: direction and magnitude. Starting from any initial point and moving in a specific direction for a specific distance brings you to the terminal point. The vector's components are found by subtracting the initial point coordinates from the terminal point coordinates.\r\n<p class=\"whitespace-normal break-words\">A vector is in <strong>standard position<\/strong> when its initial point is at the origin. When graphing vector-valued functions, we place vectors in standard position to guarantee a unique graph. This applies to both two-dimensional and three-dimensional vector-valued functions.<\/p>\r\n<p class=\"whitespace-normal break-words\">The graph of a vector-valued function traces out a curve in space. The form of the function determines what type of curve you get:<\/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>: For [latex]{\\bf{r}},(t)=f,(t),{\\bf{i}}+g,(t),{\\bf{j}}[\/latex], the graph consists of all points [latex](t,\\ {\\bf{r}},(t))[\/latex] and traces a curve in the [latex]xy[\/latex]-plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Space curve<\/strong>: For [latex]{\\bf{r}},(t)=f,(t),{\\bf{i}}+g,(t),{\\bf{j}}+h,(t),{\\bf{k}}[\/latex], the graph consists of all points [latex](t,\\ {\\bf{r}},(t))[\/latex] and traces a curve in three-dimensional space<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Any representation of a plane curve or space curve using a vector-valued function is called a <strong>vector parameterization<\/strong> of the curve.<\/p>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Vector-valued functions and parameterized curves are essentially the same thing viewed differently. 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 parameterized form. The graphs match exactly, except vector-valued functions show vectors while parameterized curves show points.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>How to: Graph a Vector-Valued Function<\/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>Create a table of values<\/strong>: Choose several values of [latex]t[\/latex] in the domain and calculate [latex]{\\bf{r}},(t)[\/latex] for each<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Plot vectors in standard position<\/strong>: Draw each vector [latex]{\\bf{r}},(t)[\/latex] with its initial point at the origin<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Mark terminal points<\/strong>: Identify where each vector ends<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Connect the points<\/strong>: Draw a smooth curve through the terminal points in order of increasing [latex]t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Indicate orientation<\/strong>: Add arrows to show the direction of the curve as [latex]t[\/latex] increases<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Create 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<\/section>\r\n<h3>Reparameterization<\/h3>\r\n<p class=\"whitespace-normal break-words\">Notice that the graphs in parts (a) and (b) are identical. This happens because the function in part (b) is a <strong>reparameterization<\/strong> of the function in part (a). Any curve has infinitely many reparameterizations. For example, replacing [latex]t[\/latex] with [latex]2t[\/latex] in any of the three curves above doesn't change the shape\u2014only the interval over which [latex]t[\/latex] is defined may change. We'll explore arc-length parameterization later in this chapter.<\/p>\r\n\r\n<h3>Helix<\/h3>\r\nThe curve in part (c) is a helix with a circular cross-section. Helices can also have elliptical cross-sections\u2014for instance, [latex]{\\bf{r}},(t)=4\\cos{t,{\\bf{i}}}+3\\sin{t,{\\bf{j}}}+t,{\\bf{k}}[\/latex] describes an elliptical helix whose projection into the [latex]xy[\/latex]-plane is an ellipse. The arrows in the graph indicate the <strong>orientation<\/strong> of the curve as [latex]t[\/latex] progresses from [latex]0[\/latex] to [latex]4\\pi[\/latex].\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>helix<\/h3>\r\nA <strong>helix<\/strong> is a three-dimensional curve that resembles a spring. When viewed along the [latex]z[\/latex]-axis, it has a circular (or elliptical) cross-section. The curve spirals upward or downward as the parameter increases.\r\n\r\n<\/section>","rendered":"<h2>Graphing Vector-Valued Functions<\/h2>\n<p>A plane vector has two components: direction and magnitude. Starting from any initial point and moving in a specific direction for a specific distance brings you to the terminal point. The vector&#8217;s components are found by subtracting the initial point coordinates from the terminal point coordinates.<\/p>\n<p class=\"whitespace-normal break-words\">A vector is in <strong>standard position<\/strong> when its initial point is at the origin. When graphing vector-valued functions, we place vectors in standard position to guarantee a unique graph. This applies to both two-dimensional and three-dimensional vector-valued functions.<\/p>\n<p class=\"whitespace-normal break-words\">The graph of a vector-valued function traces out a curve in space. The form of the function determines what type of curve you get:<\/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>: For [latex]{\\bf{r}},(t)=f,(t),{\\bf{i}}+g,(t),{\\bf{j}}[\/latex], the graph consists of all points [latex](t,\\ {\\bf{r}},(t))[\/latex] and traces a curve in the [latex]xy[\/latex]-plane<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Space curve<\/strong>: For [latex]{\\bf{r}},(t)=f,(t),{\\bf{i}}+g,(t),{\\bf{j}}+h,(t),{\\bf{k}}[\/latex], the graph consists of all points [latex](t,\\ {\\bf{r}},(t))[\/latex] and traces a curve in three-dimensional space<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Any representation of a plane curve or space curve using a vector-valued function is called a <strong>vector parameterization<\/strong> of the curve.<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Vector-valued functions and parameterized curves are essentially the same thing viewed differently. 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 parameterized form. The graphs match exactly, except vector-valued functions show vectors while parameterized curves show points.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>How to: Graph a Vector-Valued Function<\/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>Create a table of values<\/strong>: Choose several values of [latex]t[\/latex] in the domain and calculate [latex]{\\bf{r}},(t)[\/latex] for each<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Plot vectors in standard position<\/strong>: Draw each vector [latex]{\\bf{r}},(t)[\/latex] with its initial point at the origin<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Mark terminal points<\/strong>: Identify where each vector ends<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Connect the points<\/strong>: Draw a smooth curve through the terminal points in order of increasing [latex]t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Indicate orientation<\/strong>: Add arrows to show the direction of the curve as [latex]t[\/latex] increases<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794055165\">Show Solution<\/button><\/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<figure id=\"attachment_901\" aria-describedby=\"caption-attachment-901\" style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption id=\"caption-attachment-901\" class=\"wp-caption-text\">Figure 1.\u00a0The graph of the first vector-valued function is an ellipse.<\/figcaption><\/figure>\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<figure id=\"attachment_903\" aria-describedby=\"caption-attachment-903\" style=\"width: 425px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption id=\"caption-attachment-903\" class=\"wp-caption-text\">Figure 2. The graph of the second vector-valued function is also an ellipse.<\/figcaption><\/figure>\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<figure id=\"attachment_904\" aria-describedby=\"caption-attachment-904\" style=\"width: 393px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption id=\"caption-attachment-904\" class=\"wp-caption-text\">Figure 3.\u00a0The graph of the third vector-valued function is a helix.<\/figcaption><\/figure>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h3>Reparameterization<\/h3>\n<p class=\"whitespace-normal break-words\">Notice that the graphs in parts (a) and (b) are identical. This happens because the function in part (b) is a <strong>reparameterization<\/strong> of the function in part (a). Any curve has infinitely many reparameterizations. For example, replacing [latex]t[\/latex] with [latex]2t[\/latex] in any of the three curves above doesn&#8217;t change the shape\u2014only the interval over which [latex]t[\/latex] is defined may change. We&#8217;ll explore arc-length parameterization later in this chapter.<\/p>\n<h3>Helix<\/h3>\n<p>The curve in part (c) is a helix with a circular cross-section. Helices can also have elliptical cross-sections\u2014for instance, [latex]{\\bf{r}},(t)=4\\cos{t,{\\bf{i}}}+3\\sin{t,{\\bf{j}}}+t,{\\bf{k}}[\/latex] describes an elliptical helix whose projection into the [latex]xy[\/latex]-plane is an ellipse. The arrows in the graph indicate the <strong>orientation<\/strong> of the curve as [latex]t[\/latex] progresses from [latex]0[\/latex] to [latex]4\\pi[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>helix<\/h3>\n<p>A <strong>helix<\/strong> is a three-dimensional curve that resembles a spring. When viewed along the [latex]z[\/latex]-axis, it has a circular (or elliptical) cross-section. The curve spirals upward or downward as the parameter increases.<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":7,"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-948","chapter","type-chapter","status-publish","hentry"],"part":424,"module-header":"- Select Header -","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\/948","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":3,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/948\/revisions"}],"predecessor-version":[{"id":1693,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/948\/revisions\/1693"}],"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\/948\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=948"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=948"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=948"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=948"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}