{"id":40,"date":"2021-07-30T17:05:56","date_gmt":"2021-07-30T17:05:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=40"},"modified":"2022-10-20T23:57:36","modified_gmt":"2022-10-20T23:57:36","slug":"summary-of-vectors-in-the-plane","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-vectors-in-the-plane\/","title":{"raw":"Summary of Vectors in the Plane","rendered":"Summary of Vectors in the Plane"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167793944779\" data-bullet-style=\"bullet\">\r\n \t<li>Vectors are used to represent quantities that have both magnitude and direction.<\/li>\r\n \t<li>We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction.<\/li>\r\n \t<li>Subtraction of vectors is defined in terms of adding the negative of the vector.<\/li>\r\n \t<li>A vector is written in component form as\u00a0[latex]{\\bf{v}}=\\langle{x,y}\\rangle[\/latex].<\/li>\r\n \t<li>The magnitude of a vector is a scalar:\u00a0[latex]\\parallel{\\bf{v}}\\parallel=\\sqrt{x^2+y^2}[\/latex].<\/li>\r\n \t<li>A unit vector [latex]\\bf{u}[\/latex] has magnitude [latex]1[\/latex] and can be found by dividing a vector by its magnitude:\u00a0[latex]{\\bf{u}}=\\frac{1}{\\parallel{\\bf{v}}\\parallel}{\\bf{v}}[\/latex] \u00a0The standard unit vectors are\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex] and\u00a0[latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]. A vector [latex]{\\bf{v}}=\\langle{x,y}\\rangle[\/latex] can be expressed in terms of the standard unit vectors as\u00a0[latex]{\\bf{v}}=x{\\bf{i}}+y{\\bf{j}}[\/latex].<\/li>\r\n<\/ul>\r\n<div class=\"os-section-area\"><section id=\"fs-id1167794326123\" class=\"key-concepts\" data-depth=\"1\">\r\n<ul id=\"fs-id1167793944779\" data-bullet-style=\"bullet\">\r\n \t<li>Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>component<\/dt>\r\n \t<dd>a scalar that describes either the vertical or horizontal direction of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>equivalent vectors<\/dt>\r\n \t<dd>vectors that have the same magnitude and the same direction<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>initial point<\/dt>\r\n \t<dd>the starting point of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>magnitude<\/dt>\r\n \t<dd>the length of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>normalization<\/dt>\r\n \t<dd>using scalar multiplication to find a unit vector with a given direction<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>parallelogram method<\/dt>\r\n \t<dd>a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>scalar<\/dt>\r\n \t<dt><\/dt>\r\n \t<dd>a real number<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>scalar multiplication<\/dt>\r\n \t<dd>a vector operation that defines the product of a scalar and a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>standard unit vectors<\/dt>\r\n \t<dd>unit vectors along the coordinate axes:\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex], [latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>standard-position Vectors<\/dt>\r\n \t<dd>a vector with initial point [latex](0,0)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>terminal point<\/dt>\r\n \t<dd>the endpoint of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>triangle inequality<\/dt>\r\n \t<dd>the length of any side of a triangle is less than the sum of the lengths of the other two sides<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>triangle method<\/dt>\r\n \t<dd>a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>unit vector<\/dt>\r\n \t<dd>a vector with magnitude [latex]1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector<\/dt>\r\n \t<dd>a mathematical object that has both magnitude and direction<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector addition<\/dt>\r\n \t<dd>a vector operation that defines the sum of two vectors<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector difference<\/dt>\r\n \t<dd>the vector difference\u00a0[latex]{\\bf{v}}-{\\bf{w}}[\/latex]\u00a0is defined as\u00a0[latex]{\\bf{v}}+(-{\\bf{w}})={\\bf{v}}+(-1){\\bf{w}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector sum<\/dt>\r\n \t<dd>the sum of two vectors, [latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex]can be constructed graphically by placing the initial point of\u00a0[latex]{\\bf{w}}[\/latex]\u00a0at the terminal point of [latex]{\\bf{v}}[\/latex]; then the vector sum\u00a0[latex]{\\bf{v}}+{\\bf{w}}[\/latex]\u00a0is the vector with an initial point that coincides with the initial point of [latex]{\\bf{v}}[\/latex], and with a terminal point that coincides with the terminal point of\u00a0[latex]{\\bf{w}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>zero vector<\/dt>\r\n \t<dd>the vector with both initial point and terminal point\u00a0[latex](0,0)[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167793944779\" data-bullet-style=\"bullet\">\n<li>Vectors are used to represent quantities that have both magnitude and direction.<\/li>\n<li>We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction.<\/li>\n<li>Subtraction of vectors is defined in terms of adding the negative of the vector.<\/li>\n<li>A vector is written in component form as\u00a0[latex]{\\bf{v}}=\\langle{x,y}\\rangle[\/latex].<\/li>\n<li>The magnitude of a vector is a scalar:\u00a0[latex]\\parallel{\\bf{v}}\\parallel=\\sqrt{x^2+y^2}[\/latex].<\/li>\n<li>A unit vector [latex]\\bf{u}[\/latex] has magnitude [latex]1[\/latex] and can be found by dividing a vector by its magnitude:\u00a0[latex]{\\bf{u}}=\\frac{1}{\\parallel{\\bf{v}}\\parallel}{\\bf{v}}[\/latex] \u00a0The standard unit vectors are\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex] and\u00a0[latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]. A vector [latex]{\\bf{v}}=\\langle{x,y}\\rangle[\/latex] can be expressed in terms of the standard unit vectors as\u00a0[latex]{\\bf{v}}=x{\\bf{i}}+y{\\bf{j}}[\/latex].<\/li>\n<\/ul>\n<div class=\"os-section-area\">\n<section id=\"fs-id1167794326123\" class=\"key-concepts\" data-depth=\"1\">\n<ul id=\"fs-id1167793944779\" data-bullet-style=\"bullet\">\n<li>Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<\/div>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>component<\/dt>\n<dd>a scalar that describes either the vertical or horizontal direction of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>equivalent vectors<\/dt>\n<dd>vectors that have the same magnitude and the same direction<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>initial point<\/dt>\n<dd>the starting point of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>magnitude<\/dt>\n<dd>the length of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>normalization<\/dt>\n<dd>using scalar multiplication to find a unit vector with a given direction<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>parallelogram method<\/dt>\n<dd>a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>scalar<\/dt>\n<dt><\/dt>\n<dd>a real number<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>scalar multiplication<\/dt>\n<dd>a vector operation that defines the product of a scalar and a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>standard unit vectors<\/dt>\n<dd>unit vectors along the coordinate axes:\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex], [latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>standard-position Vectors<\/dt>\n<dd>a vector with initial point [latex](0,0)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>terminal point<\/dt>\n<dd>the endpoint of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>triangle inequality<\/dt>\n<dd>the length of any side of a triangle is less than the sum of the lengths of the other two sides<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>triangle method<\/dt>\n<dd>a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>unit vector<\/dt>\n<dd>a vector with magnitude [latex]1[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector<\/dt>\n<dd>a mathematical object that has both magnitude and direction<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector addition<\/dt>\n<dd>a vector operation that defines the sum of two vectors<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector difference<\/dt>\n<dd>the vector difference\u00a0[latex]{\\bf{v}}-{\\bf{w}}[\/latex]\u00a0is defined as\u00a0[latex]{\\bf{v}}+(-{\\bf{w}})={\\bf{v}}+(-1){\\bf{w}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector sum<\/dt>\n<dd>the sum of two vectors, [latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex]can be constructed graphically by placing the initial point of\u00a0[latex]{\\bf{w}}[\/latex]\u00a0at the terminal point of [latex]{\\bf{v}}[\/latex]; then the vector sum\u00a0[latex]{\\bf{v}}+{\\bf{w}}[\/latex]\u00a0is the vector with an initial point that coincides with the initial point of [latex]{\\bf{v}}[\/latex], and with a terminal point that coincides with the terminal point of\u00a0[latex]{\\bf{w}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>zero vector<\/dt>\n<dd>the vector with both initial point and terminal point\u00a0[latex](0,0)[\/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-40\">\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":6,"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-40","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/40","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":12,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/40\/revisions"}],"predecessor-version":[{"id":3685,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/40\/revisions\/3685"}],"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\/40\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=40"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=40"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=40"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=40"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}