{"id":44,"date":"2021-07-30T17:06:01","date_gmt":"2021-07-30T17:06:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=44"},"modified":"2022-10-21T00:15:21","modified_gmt":"2022-10-21T00:15:21","slug":"summary-of-the-dot-product","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-the-dot-product\/","title":{"raw":"Summary of the Dot Product","rendered":"Summary of the Dot Product"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The dot product, or scalar product, of two vectors\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex] is\u00a0[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex].<\/li>\r\n \t<li>The dot product satisfies the following properties:\r\n<ul>\r\n \t<li>[latex]{\\bf{u}}\\cdot{\\bf{v}}={\\bf{v}}\\cdot{\\bf{u}}[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{u}}\\cdot\\left({\\bf{v}}+{\\bf{w}}\\right)={\\bf{u}}\\cdot{\\bf{v}}+{\\bf{u}}\\cdot{\\bf{w}}[\/latex]<\/li>\r\n \t<li>[latex]c({\\bf{u}}\\cdot{\\bf{v}})=(c{\\bf{u}})\\cdot{\\bf{v}}={\\bf{u}}\\cdot(c{\\bf{v}})[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{v}}\\cdot{\\bf{v}}=\\parallel{\\bf{v}}\\parallel^{2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The dot product of two vectors can be expressed, alternatively, as [latex]{\\bf{u}}\\cdot{\\bf{v}}=\\parallel{\\bf{u}}\\parallel\\parallel{\\bf{v}}\\parallel\\cos\\theta[\/latex].\u00a0This form of the dot product is useful for finding the measure of the angle formed by two vectors.<\/li>\r\n \t<li>Vectors\u00a0[latex]\\bf{u}[\/latex]\u00a0and\u00a0[latex]\\bf{v}[\/latex]\u00a0are orthogonal if [latex]{\\bf{u}}\\cdot{\\bf{v}}=0[\/latex].<\/li>\r\n \t<li>The angles formed by a nonzero vector and the coordinate axes are called the\u00a0<em data-effect=\"italics\">direction angles<\/em>\u00a0for the vector. The cosines of these angles are known as the\u00a0<em data-effect=\"italics\">direction cosines<\/em>.<\/li>\r\n \t<li>The vector projection of\u00a0[latex]\\bf{v}[\/latex]\u00a0onto\u00a0[latex]\\bf{u}[\/latex] is the vector [latex]\\text{proj}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel^{2}}{\\bf{u}}[\/latex].\u00a0The magnitude of this vector is known as the\u00a0<em data-effect=\"italics\">scalar projection<\/em>\u00a0of [latex]\\bf{v}[\/latex]\u00a0onto\u00a0[latex]\\bf{u}[\/latex], given by\u00a0[latex]\\text{comp}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel}[\/latex].<\/li>\r\n \t<li>\r\n<div class=\"os-section-area\"><section id=\"fs-id1163723268353\" class=\"key-concepts\" data-depth=\"1\">Work is done when a force is applied to an object, causing displacement. When the force is represented by the vector [latex]\\bf{F}[\/latex]\u00a0and the displacement is represented by the vector\u00a0[latex]\\bf{s}[\/latex], then the work done [latex]W[\/latex]\u00a0is given by the formula [latex]W={\\bf{F}}\\cdot{\\bf{s}}=\\parallel{\\bf{F}}\\parallel\\parallel{\\bf{s}}\\parallel\\cos\\theta[\/latex].<\/section><\/div><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Dot product of u and v\r\n<\/strong>[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex]<\/li>\r\n \t<li><strong>Cosine of the angle formed by u and v<\/strong>\r\n[latex]\\cos\\theta=\\frac{{\\bf{u}}\\cdot{\\bf{v}}}{\\parallel{\\bf{u}}\\parallel\\parallel{\\bf{v}}\\parallel}[\/latex]<\/li>\r\n \t<li><strong>Vector projection of v onto u<\/strong>\r\n[latex]\\text{proj}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel^{2}}{\\bf{u}}[\/latex]<\/li>\r\n \t<li><strong>Scalar projection of v onto u<\/strong>\r\n[latex]\\text{comp}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel}[\/latex]<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Work done by a force F to move an object through displacement vector [latex]\\overrightarrow{PQ}[\/latex]<\/strong>\r\n[latex]W={\\bf{F}}\\cdot{\\bf{s}}=\\parallel{\\bf{F}}\\parallel\\parallel{\\bf{s}}\\parallel\\cos\\theta[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>direction angles<\/dt>\r\n \t<dd>the angles formed by a nonzero vector and the coordinate axes<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>direction cosines<\/dt>\r\n \t<dd>the cosines of the angles formed by a nonzero vector and the coordinate axes<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>dot product or scalar product<\/dt>\r\n \t<dd>[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex], where\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex]\u00a0and[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>orthogonal vectors<\/dt>\r\n \t<dd>vectors that form a right angle when placed in standard position<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>scalar projection<\/dt>\r\n \t<dd>the magnitude of the vector projection of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector projection<\/dt>\r\n \t<dd>the component of a vector that follows a given direction<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>work done by a force<\/dt>\r\n \t<dd>work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector [latex]\\bf{F}[\/latex] and the displacement of an object by a vector [latex]\\bf{s}[\/latex], then the work done by the force is the dot product of [latex]\\bf{F}[\/latex]\u00a0and [latex]\\bf{s}[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The dot product, or scalar product, of two vectors\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex] is\u00a0[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex].<\/li>\n<li>The dot product satisfies the following properties:\n<ul>\n<li>[latex]{\\bf{u}}\\cdot{\\bf{v}}={\\bf{v}}\\cdot{\\bf{u}}[\/latex]<\/li>\n<li>[latex]{\\bf{u}}\\cdot\\left({\\bf{v}}+{\\bf{w}}\\right)={\\bf{u}}\\cdot{\\bf{v}}+{\\bf{u}}\\cdot{\\bf{w}}[\/latex]<\/li>\n<li>[latex]c({\\bf{u}}\\cdot{\\bf{v}})=(c{\\bf{u}})\\cdot{\\bf{v}}={\\bf{u}}\\cdot(c{\\bf{v}})[\/latex]<\/li>\n<li>[latex]{\\bf{v}}\\cdot{\\bf{v}}=\\parallel{\\bf{v}}\\parallel^{2}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>The dot product of two vectors can be expressed, alternatively, as [latex]{\\bf{u}}\\cdot{\\bf{v}}=\\parallel{\\bf{u}}\\parallel\\parallel{\\bf{v}}\\parallel\\cos\\theta[\/latex].\u00a0This form of the dot product is useful for finding the measure of the angle formed by two vectors.<\/li>\n<li>Vectors\u00a0[latex]\\bf{u}[\/latex]\u00a0and\u00a0[latex]\\bf{v}[\/latex]\u00a0are orthogonal if [latex]{\\bf{u}}\\cdot{\\bf{v}}=0[\/latex].<\/li>\n<li>The angles formed by a nonzero vector and the coordinate axes are called the\u00a0<em data-effect=\"italics\">direction angles<\/em>\u00a0for the vector. The cosines of these angles are known as the\u00a0<em data-effect=\"italics\">direction cosines<\/em>.<\/li>\n<li>The vector projection of\u00a0[latex]\\bf{v}[\/latex]\u00a0onto\u00a0[latex]\\bf{u}[\/latex] is the vector [latex]\\text{proj}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel^{2}}{\\bf{u}}[\/latex].\u00a0The magnitude of this vector is known as the\u00a0<em data-effect=\"italics\">scalar projection<\/em>\u00a0of [latex]\\bf{v}[\/latex]\u00a0onto\u00a0[latex]\\bf{u}[\/latex], given by\u00a0[latex]\\text{comp}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel}[\/latex].<\/li>\n<li>\n<div class=\"os-section-area\">\n<section id=\"fs-id1163723268353\" class=\"key-concepts\" data-depth=\"1\">Work is done when a force is applied to an object, causing displacement. When the force is represented by the vector [latex]\\bf{F}[\/latex]\u00a0and the displacement is represented by the vector\u00a0[latex]\\bf{s}[\/latex], then the work done [latex]W[\/latex]\u00a0is given by the formula [latex]W={\\bf{F}}\\cdot{\\bf{s}}=\\parallel{\\bf{F}}\\parallel\\parallel{\\bf{s}}\\parallel\\cos\\theta[\/latex].<\/section>\n<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Dot product of u and v<br \/>\n<\/strong>[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex]<\/li>\n<li><strong>Cosine of the angle formed by u and v<\/strong><br \/>\n[latex]\\cos\\theta=\\frac{{\\bf{u}}\\cdot{\\bf{v}}}{\\parallel{\\bf{u}}\\parallel\\parallel{\\bf{v}}\\parallel}[\/latex]<\/li>\n<li><strong>Vector projection of v onto u<\/strong><br \/>\n[latex]\\text{proj}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel^{2}}{\\bf{u}}[\/latex]<\/li>\n<li><strong>Scalar projection of v onto u<\/strong><br \/>\n[latex]\\text{comp}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel}[\/latex]<\/li>\n<\/ul>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Work done by a force F to move an object through displacement vector [latex]\\overrightarrow{PQ}[\/latex]<\/strong><br \/>\n[latex]W={\\bf{F}}\\cdot{\\bf{s}}=\\parallel{\\bf{F}}\\parallel\\parallel{\\bf{s}}\\parallel\\cos\\theta[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>direction angles<\/dt>\n<dd>the angles formed by a nonzero vector and the coordinate axes<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>direction cosines<\/dt>\n<dd>the cosines of the angles formed by a nonzero vector and the coordinate axes<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>dot product or scalar product<\/dt>\n<dd>[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex], where\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex]\u00a0and[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>orthogonal vectors<\/dt>\n<dd>vectors that form a right angle when placed in standard position<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>scalar projection<\/dt>\n<dd>the magnitude of the vector projection of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector projection<\/dt>\n<dd>the component of a vector that follows a given direction<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>work done by a force<\/dt>\n<dd>work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector [latex]\\bf{F}[\/latex] and the displacement of an object by a vector [latex]\\bf{s}[\/latex], then the work done by the force is the dot product of [latex]\\bf{F}[\/latex]\u00a0and [latex]\\bf{s}[\/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-44\">\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":15,"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\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-44","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/44","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":14,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions"}],"predecessor-version":[{"id":3687,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions\/3687"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/44\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=44"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=44"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=44"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=44"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}