{"id":68,"date":"2021-07-30T17:08:22","date_gmt":"2021-07-30T17:08:22","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=68"},"modified":"2022-10-29T00:33:11","modified_gmt":"2022-10-29T00:33:11","slug":"summary-of-vector-valued-functions-and-space-curves","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-vector-valued-functions-and-space-curves\/","title":{"raw":"Summary of Vector-Valued Functions and Space Curves","rendered":"Summary of Vector-Valued Functions and Space Curves"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>A vector-valued function is a function of the form\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] or\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex]\u00a0where the component functions [latex]f[\/latex], [latex]g[\/latex], and\u00a0[latex]h[\/latex] are real-valued functions of the parameter\u00a0[latex]t[\/latex].<\/li>\r\n \t<li>The graph of a vector-valued function of the form\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex]\u00a0is called a\u00a0<em data-effect=\"italics\">plane curve<\/em>. The graph of a vector-valued function of the form\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex]\u00a0is called a\u00a0<em data-effect=\"italics\">space curve<\/em>.<\/li>\r\n \t<li>It is possible to represent an arbitrary plane curve by a vector-valued function.<\/li>\r\n \t<li>To calculate the limit of a vector-valued function, calculate the limits of the component functions separately.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Vector-valued function\r\n<\/strong>[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] or\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex], or\u00a0[latex]{\\bf{r}}(t)=\\langle{f(t),g(t)}\\rangle[\/latex] or\u00a0[latex]{\\bf{r}}(t)=\\langle{f(t),g(t),h(t)}\\rangle[\/latex]<\/li>\r\n \t<li><strong>Limit of a vector-valued function<\/strong>\r\n[latex]\\underset{t\\to{a}}{\\lim}{\\bf{r}}(t)=\\left[\\underset{t\\to{a}}{\\lim}f(t)\\right]{\\bf{i}}+\\left[\\underset{t\\to{a}}{\\lim}g(t)\\right]{\\bf{j}}[\/latex] or [latex]\\underset{t\\to{a}}{\\lim}{\\bf{r}}(t)=\\left[\\underset{t\\to{a}}{\\lim}f(t)\\right]{\\bf{i}}+\\left[\\underset{t\\to{a}}{\\lim}g(t)\\right]{\\bf{j}}+\\left[\\underset{t\\to{a}}{\\lim}h(t)\\right]{\\bf{k}}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>component functions<\/dt>\r\n \t<dd>the component functions of the vector-valued function [latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] are [latex]f(t)[\/latex] and [latex]g(t)[\/latex], and the component functions of the vector-valued function\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex] are\u00a0[latex]f(t)[\/latex],\u00a0[latex]g(t)[\/latex] and [latex]h(t)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>helix<\/dt>\r\n \t<dd>a three-dimensional curve in the shape of a spiral<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>limit of a vector-valued function<\/dt>\r\n \t<dd>a vector-valued function [latex]{\\bf{r}}(t)[\/latex] has a limit [latex]{\\bf{L}}[\/latex] as [latex]t[\/latex]<em>\u00a0<\/em>approaches\u00a0[latex]a[\/latex]\u00a0if [latex]\\underset{t\\to{a}}{\\lim}|{\\bf{r}}(t)-{\\bf{L}}|=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>plane curve<\/dt>\r\n \t<dd>the set of ordered pairs [latex]\\left(f(t),g(t)\\right)[\/latex] together with their defining parametric equations\u00a0[latex]x=f(t)[\/latex] and [latex]y=g(t)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>reparameterization<\/dt>\r\n \t<dd>an alternative parameterization of a given vector-valued function<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>space curve<\/dt>\r\n \t<dd>the set of ordered triples\u00a0[latex]\\left(f(t),g(t),h(t)\\right)[\/latex]\u00a0together with their defining parametric equations [latex]x=f(t)[\/latex],\u00a0[latex]y=g(t)[\/latex] and [latex]z=h(t)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector parameterization<\/dt>\r\n \t<dd>any representation of a plane or space curve using a vector-valued function<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector-valued function<\/dt>\r\n \t<dd>a function of the form [latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] or\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex],<strong>\u00a0<\/strong>where the component functions [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are real-valued functions of the parameter [latex]t[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>A vector-valued function is a function of the form\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] or\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex]\u00a0where the component functions [latex]f[\/latex], [latex]g[\/latex], and\u00a0[latex]h[\/latex] are real-valued functions of the parameter\u00a0[latex]t[\/latex].<\/li>\n<li>The graph of a vector-valued function of the form\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex]\u00a0is called a\u00a0<em data-effect=\"italics\">plane curve<\/em>. The graph of a vector-valued function of the form\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex]\u00a0is called a\u00a0<em data-effect=\"italics\">space curve<\/em>.<\/li>\n<li>It is possible to represent an arbitrary plane curve by a vector-valued function.<\/li>\n<li>To calculate the limit of a vector-valued function, calculate the limits of the component functions separately.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Vector-valued function<br \/>\n<\/strong>[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] or\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex], or\u00a0[latex]{\\bf{r}}(t)=\\langle{f(t),g(t)}\\rangle[\/latex] or\u00a0[latex]{\\bf{r}}(t)=\\langle{f(t),g(t),h(t)}\\rangle[\/latex]<\/li>\n<li><strong>Limit of a vector-valued function<\/strong><br \/>\n[latex]\\underset{t\\to{a}}{\\lim}{\\bf{r}}(t)=\\left[\\underset{t\\to{a}}{\\lim}f(t)\\right]{\\bf{i}}+\\left[\\underset{t\\to{a}}{\\lim}g(t)\\right]{\\bf{j}}[\/latex] or [latex]\\underset{t\\to{a}}{\\lim}{\\bf{r}}(t)=\\left[\\underset{t\\to{a}}{\\lim}f(t)\\right]{\\bf{i}}+\\left[\\underset{t\\to{a}}{\\lim}g(t)\\right]{\\bf{j}}+\\left[\\underset{t\\to{a}}{\\lim}h(t)\\right]{\\bf{k}}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>component functions<\/dt>\n<dd>the component functions of the vector-valued function [latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] are [latex]f(t)[\/latex] and [latex]g(t)[\/latex], and the component functions of the vector-valued function\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex] are\u00a0[latex]f(t)[\/latex],\u00a0[latex]g(t)[\/latex] and [latex]h(t)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>helix<\/dt>\n<dd>a three-dimensional curve in the shape of a spiral<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>limit of a vector-valued function<\/dt>\n<dd>a vector-valued function [latex]{\\bf{r}}(t)[\/latex] has a limit [latex]{\\bf{L}}[\/latex] as [latex]t[\/latex]<em>\u00a0<\/em>approaches\u00a0[latex]a[\/latex]\u00a0if [latex]\\underset{t\\to{a}}{\\lim}|{\\bf{r}}(t)-{\\bf{L}}|=0[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>plane curve<\/dt>\n<dd>the set of ordered pairs [latex]\\left(f(t),g(t)\\right)[\/latex] together with their defining parametric equations\u00a0[latex]x=f(t)[\/latex] and [latex]y=g(t)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>reparameterization<\/dt>\n<dd>an alternative parameterization of a given vector-valued function<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>space curve<\/dt>\n<dd>the set of ordered triples\u00a0[latex]\\left(f(t),g(t),h(t)\\right)[\/latex]\u00a0together with their defining parametric equations [latex]x=f(t)[\/latex],\u00a0[latex]y=g(t)[\/latex] and [latex]z=h(t)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector parameterization<\/dt>\n<dd>any representation of a plane or space curve using a vector-valued function<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector-valued function<\/dt>\n<dd>a function of the form [latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] or\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex],<strong>\u00a0<\/strong>where the component functions [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are real-valued functions of the parameter [latex]t[\/latex]<\/dd>\n<\/dl>\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-68\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><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":17533,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) 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https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-68","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/68","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\/17533"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/68\/revisions"}],"predecessor-version":[{"id":3693,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/68\/revisions\/3693"}],"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\/68\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=68"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=68"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=68"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=68"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}