{"id":42,"date":"2021-07-30T17:05:59","date_gmt":"2021-07-30T17:05:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=42"},"modified":"2022-10-21T00:03:42","modified_gmt":"2022-10-21T00:03:42","slug":"summary-of-vectors-in-three-dimensions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-vectors-in-three-dimensions\/","title":{"raw":"Summary of Vectors in Three Dimensions","rendered":"Summary of Vectors in Three Dimensions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin.\u00a0Ordered triples [latex](x,y,z)[\/latex]\u00a0are used to describe the location of a point in space.<\/li>\r\n \t<li>The distance [latex]d[\/latex]\u00a0between points [latex](x_{1},y_{1},z_{1})[\/latex] and\u00a0[latex](x_{2},y_{2},z_{2})[\/latex]\u00a0is given by the formula [latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[\/latex].<\/li>\r\n \t<li>In three dimensions, the equations [latex]x=a[\/latex],\u00a0[latex]y=b[\/latex], and\u00a0[latex]z=c[\/latex]\u00a0describe planes that are parallel to the coordinate planes.<\/li>\r\n \t<li>The standard equation of a sphere with center [latex](a,b,c)[\/latex] and radius [latex]r[\/latex]\u00a0is [latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex].<\/li>\r\n \t<li>In three dimensions, as in two, vectors are commonly expressed in component form, [latex]{\\bf{v}}=\\langle{x,y,z}\\rangle[\/latex], or in terms of the standard unit vectors,\u00a0[latex]x{\\bf{i}}+y{\\bf{j}}+z{\\bf{k}}[\/latex].<\/li>\r\n \t<li>Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let\u00a0[latex]{\\bf{v}}=\\langle{x_{1},y_{1},z_{1}}\\rangle[\/latex] and\u00a0[latex]{\\bf{w}}=\\langle{x_{2},y_{2},z_{2}}\\rangle[\/latex]\u00a0be vectors, and let [latex]k[\/latex]\u00a0be a scalar.\r\n<ul>\r\n \t<li>Scalar multiplication:\u00a0[latex]k{\\bf{v}}=\\langle{kx_{1},ky_{1},kz_{1}}\\rangle[\/latex]<\/li>\r\n \t<li>Vector addition:\u00a0[latex]{\\bf{v}}+{\\bf{w}}=\\langle{x_{1},y_{1},z_{1}}\\rangle+\\langle{x_{2},y_{2},z_{2}}\\rangle=\\langle{x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2}}\\rangle[\/latex]<\/li>\r\n \t<li>Vector subtraction:\u00a0[latex]{\\bf{v}}-{\\bf{w}}=\\langle{x_{1},y_{1},z_{1}}\\rangle-\\langle{x_{2},y_{2},z_{2}}\\rangle=\\langle{x_{1}-x_{2},y_{1}-y_{2},z_{1}-z_{2}}\\rangle[\/latex]<\/li>\r\n \t<li>Vector magnitude: [latex]\\parallel{\\bf{v}}\\parallel=\\sqrt{{x_{1}}^{2}+{y_{1}}^{2}+{z_{1}}^{2}}[\/latex]<\/li>\r\n \t<li>Unit vector in the direction of [latex]{\\bf{v}}[\/latex]: [latex]\\frac{\\bf{v}}{\\parallel{\\bf{v}}\\parallel}=\\frac{1}{\\parallel{\\bf{v}}\\parallel}\\langle{x_{1},y_{1},z_{1}}\\rangle=\\langle{\\frac{x_{1}}{\\parallel{\\bf{v}}\\parallel},\\frac{y_{1}}{\\parallel{\\bf{v}}\\parallel},\\frac{z_{1}}{\\parallel{\\bf{v}}\\parallel}}\\rangle,{\\bf{v}}\\ne{0}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169736654969\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Distance between two points in space<\/strong>\r\n[latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[\/latex]<\/li>\r\n \t<li><strong>Sphere with center [latex](a,b,c)[\/latex] and radius [latex]r[\/latex]<\/strong>\r\n[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>coordinate plane<\/dt>\r\n \t<dd>a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the [latex]xy[\/latex]-plane, [latex]xz[\/latex]-plane, or the [latex]yz[\/latex]<span style=\"font-size: 1em;\">-plane<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>octants<\/dt>\r\n \t<dd>the eight regions of space created by the coordinate planes<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>right-hand rule<\/dt>\r\n \t<dd>a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the [latex]z[\/latex]<span style=\"font-size: 1em;\">-axis in such a way that the fingers curl from the positive [latex]x[\/latex]-axis to the positive [latex]y[\/latex]-axis, the thumb points in the direction of the positive [latex]z[\/latex]-axis<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>sphere<\/dt>\r\n \t<dd>the set of all points equidistant from a given point known as the <em style=\"font-size: 1em;\" data-effect=\"italics\">center<\/em><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>standard equation of a sphere<\/dt>\r\n \t<dd>[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex] describes a sphere with center [latex](a,b,c)[\/latex]<span class=\"Apple-converted-space\"><span style=\"color: #000000;\"><span style=\"caret-color: #000000; font-size: 14px; white-space: nowrap;\">\u00a0and radius [latex]r[\/latex]<\/span><\/span><\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>three-dimensional rectangular coordinate system<\/dt>\r\n \t<dd>a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple\u00a0[latex](x,y,z)[\/latex]\u00a0that plots\u00a0its<span style=\"font-size: 1em;\">\u00a0location relative to the defining axes<\/span><\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin.\u00a0Ordered triples [latex](x,y,z)[\/latex]\u00a0are used to describe the location of a point in space.<\/li>\n<li>The distance [latex]d[\/latex]\u00a0between points [latex](x_{1},y_{1},z_{1})[\/latex] and\u00a0[latex](x_{2},y_{2},z_{2})[\/latex]\u00a0is given by the formula [latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[\/latex].<\/li>\n<li>In three dimensions, the equations [latex]x=a[\/latex],\u00a0[latex]y=b[\/latex], and\u00a0[latex]z=c[\/latex]\u00a0describe planes that are parallel to the coordinate planes.<\/li>\n<li>The standard equation of a sphere with center [latex](a,b,c)[\/latex] and radius [latex]r[\/latex]\u00a0is [latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex].<\/li>\n<li>In three dimensions, as in two, vectors are commonly expressed in component form, [latex]{\\bf{v}}=\\langle{x,y,z}\\rangle[\/latex], or in terms of the standard unit vectors,\u00a0[latex]x{\\bf{i}}+y{\\bf{j}}+z{\\bf{k}}[\/latex].<\/li>\n<li>Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let\u00a0[latex]{\\bf{v}}=\\langle{x_{1},y_{1},z_{1}}\\rangle[\/latex] and\u00a0[latex]{\\bf{w}}=\\langle{x_{2},y_{2},z_{2}}\\rangle[\/latex]\u00a0be vectors, and let [latex]k[\/latex]\u00a0be a scalar.\n<ul>\n<li>Scalar multiplication:\u00a0[latex]k{\\bf{v}}=\\langle{kx_{1},ky_{1},kz_{1}}\\rangle[\/latex]<\/li>\n<li>Vector addition:\u00a0[latex]{\\bf{v}}+{\\bf{w}}=\\langle{x_{1},y_{1},z_{1}}\\rangle+\\langle{x_{2},y_{2},z_{2}}\\rangle=\\langle{x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2}}\\rangle[\/latex]<\/li>\n<li>Vector subtraction:\u00a0[latex]{\\bf{v}}-{\\bf{w}}=\\langle{x_{1},y_{1},z_{1}}\\rangle-\\langle{x_{2},y_{2},z_{2}}\\rangle=\\langle{x_{1}-x_{2},y_{1}-y_{2},z_{1}-z_{2}}\\rangle[\/latex]<\/li>\n<li>Vector magnitude: [latex]\\parallel{\\bf{v}}\\parallel=\\sqrt{{x_{1}}^{2}+{y_{1}}^{2}+{z_{1}}^{2}}[\/latex]<\/li>\n<li>Unit vector in the direction of [latex]{\\bf{v}}[\/latex]: [latex]\\frac{\\bf{v}}{\\parallel{\\bf{v}}\\parallel}=\\frac{1}{\\parallel{\\bf{v}}\\parallel}\\langle{x_{1},y_{1},z_{1}}\\rangle=\\langle{\\frac{x_{1}}{\\parallel{\\bf{v}}\\parallel},\\frac{y_{1}}{\\parallel{\\bf{v}}\\parallel},\\frac{z_{1}}{\\parallel{\\bf{v}}\\parallel}}\\rangle,{\\bf{v}}\\ne{0}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169736654969\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Distance between two points in space<\/strong><br \/>\n[latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[\/latex]<\/li>\n<li><strong>Sphere with center [latex](a,b,c)[\/latex] and radius [latex]r[\/latex]<\/strong><br \/>\n[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>coordinate plane<\/dt>\n<dd>a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the [latex]xy[\/latex]-plane, [latex]xz[\/latex]-plane, or the [latex]yz[\/latex]<span style=\"font-size: 1em;\">-plane<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>octants<\/dt>\n<dd>the eight regions of space created by the coordinate planes<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>right-hand rule<\/dt>\n<dd>a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the [latex]z[\/latex]<span style=\"font-size: 1em;\">-axis in such a way that the fingers curl from the positive [latex]x[\/latex]-axis to the positive [latex]y[\/latex]-axis, the thumb points in the direction of the positive [latex]z[\/latex]-axis<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>sphere<\/dt>\n<dd>the set of all points equidistant from a given point known as the <em style=\"font-size: 1em;\" data-effect=\"italics\">center<\/em><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>standard equation of a sphere<\/dt>\n<dd>[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex] describes a sphere with center [latex](a,b,c)[\/latex]<span class=\"Apple-converted-space\"><span style=\"color: #000000;\"><span style=\"caret-color: #000000; font-size: 14px; white-space: nowrap;\">\u00a0and radius [latex]r[\/latex]<\/span><\/span><\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>three-dimensional rectangular coordinate system<\/dt>\n<dd>a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple\u00a0[latex](x,y,z)[\/latex]\u00a0that plots\u00a0its<span style=\"font-size: 1em;\">\u00a0location relative to the defining axes<\/span><\/dd>\n<\/dl>\n<\/div>\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-42\">\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":11,"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-42","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/42","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":15,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/42\/revisions"}],"predecessor-version":[{"id":3686,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/42\/revisions\/3686"}],"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\/42\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=42"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=42"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=42"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=42"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}