{"id":17982,"date":"2021-09-11T20:08:20","date_gmt":"2021-09-11T20:08:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=17982"},"modified":"2021-09-11T20:43:31","modified_gmt":"2021-09-11T20:43:31","slug":"section-9-5-the-dot-product","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/section-9-5-the-dot-product\/","title":{"raw":"Section 9.5: The Dot Product","rendered":"Section 9.5: The Dot Product"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the dot product of two vectors.<\/li>\r\n \t<li>Find the angle between two vectors.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>\u00a0Finding the Dot Product of Two Vectors<\/h2>\r\nAs we discussed in the previous section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the <em>dot product<\/em> and the <em>cross product<\/em>. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.\r\n\r\nThe dot product of two vectors involves multiplying two vectors together, and the result is a scalar.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Dot Product<\/h3>\r\nThe <strong>dot product<\/strong> of two vectors [latex]\\boldsymbol{v}=\\langle a,b\\rangle [\/latex] and [latex]\\boldsymbol{v}=\\langle c,d\\rangle [\/latex] is the sum of the product of the horizontal components and the product of the vertical components.\r\n<p style=\"text-align: center;\">[latex]\\boldsymbol{v}\\cdot \\boldsymbol{u}=ac+bd[\/latex]<\/p>\r\nTo find the angle between the two vectors, use the formula below.\r\n<p style=\"text-align: center;\">[latex]\\cos \\theta =\\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Finding the Dot Product of Two Vectors<\/h3>\r\nFind the dot product of [latex]\\boldsymbol{v}=\\langle 5,12\\rangle [\/latex] and [latex]\\boldsymbol{u}=\\langle -3,4\\rangle [\/latex].\r\n\r\n[reveal-answer q=\"734442\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"734442\"]\r\n\r\nUsing the formula, we have\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{v}\\cdot \\boldsymbol{u}&amp;=\\langle 5,12\\rangle \\cdot \\langle -3,4\\rangle \\\\ &amp;=5\\cdot \\left(-3\\right)+12\\cdot 4 \\\\ &amp;=-15+48 \\\\ &amp;=33 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Finding the Dot Product of Two Vectors and the Angle between Them<\/h3>\r\nFind the dot product of <strong><em>v<\/em><\/strong><sub>1<\/sub> = 5<strong><em>i<\/em><\/strong> + 2<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong><sub>2<\/sub> = 3<strong><em>i<\/em><\/strong> + 7<strong><em>j<\/em><\/strong>. Then, find the angle between the two vectors.\r\n\r\n[reveal-answer q=\"817790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"817790\"]\r\n\r\nFinding the dot product, we multiply corresponding components.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{\\boldsymbol{v}}_{1}\\cdot {\\boldsymbol{v}}_{2}&amp;=\\langle 5,2\\rangle \\cdot \\langle 3,7\\rangle \\\\ &amp;=5\\cdot 3+2\\cdot 7 \\\\ &amp;=15+14 \\\\ &amp;=29 \\end{align}[\/latex]<\/p>\r\nTo find the angle between them, we use the formula [latex]\\cos \\theta =\\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\frac{\\boldsymbol{u}}{|\\boldsymbol{u}|}&amp;=\\langle \\frac{5}{\\sqrt{29}}+\\frac{2}{\\sqrt{29}}\\rangle \\cdot \\langle \\frac{3}{\\sqrt{58}}+\\frac{7}{\\sqrt{58}}\\rangle \\\\ &amp;=\\frac{5}{\\sqrt{29}}\\cdot \\frac{3}{\\sqrt{58}}+\\frac{2}{\\sqrt{29}}\\cdot \\frac{7}{\\sqrt{58}} \\\\ &amp;=\\frac{15}{\\sqrt{1682}}+\\frac{14}{\\sqrt{1682}}=\\frac{29}{\\sqrt{1682}} \\\\ &amp;=0.707107 \\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\theta={\\cos }^{-1}\\left(0.707107\\right)=45^\\circ[\/latex]<\/p>\r\n\r\n<div class=\"mceTemp\"><\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181206\/CNX_Precalc_Figure_08_08_0142.jpg\" alt=\"Plot showing the two position vectors (3,7) and (5,2) and the 45 degree angle between them.\" width=\"487\" height=\"403\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Finding the Angle between Two Vectors<\/h3>\r\nFind the angle between [latex]\\boldsymbol{u}=\\langle -3,4\\rangle [\/latex] and [latex]\\boldsymbol{v}=\\langle 5,12\\rangle [\/latex].\r\n\r\n[reveal-answer q=\"953383\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"953383\"]\r\n\r\nUsing the formula, [latex]\\theta ={\\cos }^{-1}\\left(\\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}\\cdot \\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\right)[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\begin{align} \\left(\\frac{\\boldsymbol{u}}{|\\boldsymbol{u}|}\\cdot \\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\right)&amp;=\\frac{-3\\boldsymbol{i}+4\\boldsymbol{j}}{5}\\cdot \\frac{5\\boldsymbol{i}+12\\boldsymbol{j}}{13} \\\\ &amp;=\\left(-\\frac{3}{5}\\cdot \\frac{5}{13}\\right)+\\left(\\frac{4}{5}\\cdot \\frac{12}{13}\\right) \\\\ &amp;=-\\frac{15}{65}+\\frac{48}{65} \\\\ &amp;=\\frac{33}{65}\\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\theta ={\\cos }^{-1}\\left(\\frac{33}{65}\\right) ={59.5}^{\\circ }[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><img style=\"font-weight: bold; background-color: #f5f5f5; font-size: 0.9em; text-align: left;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181209\/CNX_Precalc_Figure_08_08_0132.jpg\" alt=\"Plot showing the two position vectors (-3,4) and (5,12) and the 59.5 degree angle between them.\" width=\"487\" height=\"628\" \/><\/p>\r\n<b>Figure 2<\/b>[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the dot product of [latex]\\boldsymbol{u}=[\/latex]\u00a04<strong><em>i<\/em><\/strong> - 3<strong><em>j<\/em><\/strong> and [latex]\\boldsymbol{v}=[\/latex]\u00a02<strong><em>i<\/em><\/strong> + 5<strong><em>j<\/em><\/strong>. Then, find the angle between the two vectors.\r\n\r\n[reveal-answer q=\"453643\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"453643\"]\r\n\r\n[latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}=-7[\/latex]\r\n\r\n[latex]\\theta\\approx105^\\circ[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>The dot product of two vectors is the product of the<em><strong> i <\/strong><\/em>terms plus the product of the<em><strong> j <\/strong><\/em>terms.<\/li>\r\n \t<li>We can use the dot product to find the angle between two vectors.<\/li>\r\n \t<li>Dot products are useful for many types of physics applications.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165133447852\" class=\"definition\">\r\n \t<dt>dot product<\/dt>\r\n \t<dd id=\"fs-id1165133447857\">given two vectors, the sum of the product of the horizontal components and the product of the vertical components<\/dd>\r\n \t<dd><\/dd>\r\n<\/dl>\r\n<h2 style=\"text-align: center;\">Section 9.5 Homework Exercises<\/h2>\r\n1.\u00a0Given [latex]\\boldsymbol{u}=\\boldsymbol{i}\u2212\\boldsymbol{j}[\/latex] and [latex]\\boldsymbol{v}=\\boldsymbol{i}+\\boldsymbol{j}[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.\r\n\r\n2.\u00a0Given [latex]\\boldsymbol{u}=3\\boldsymbol{i}\u22124\\boldsymbol{j}[\/latex] and [latex]\\boldsymbol{v}=\u22122\\boldsymbol{i}+3\\boldsymbol{j}[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.\r\n\r\n3.\u00a0Given [latex]\\boldsymbol{u}=\u2212\\boldsymbol{i}\u2212\\boldsymbol{j}[\/latex] and [latex]\\boldsymbol{v}=\\boldsymbol{i}+5\\boldsymbol{j}[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.\r\n\r\n4.\u00a0Given [latex]\\boldsymbol{u}=\\langle\u22122,4\\rangle[\/latex] and [latex]\\boldsymbol{v}=\\langle\u22123,1\\rangle[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.\r\n\r\n5.\u00a0Given [latex]\\boldsymbol{u}=\\langle\u22121,6\\rangle[\/latex] and [latex]\\boldsymbol{v}=\\langle 6,\u22121\\rangle[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.\r\n\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the dot product of two vectors.<\/li>\n<li>Find the angle between two vectors.<\/li>\n<\/ul>\n<\/div>\n<h2>\u00a0Finding the Dot Product of Two Vectors<\/h2>\n<p>As we discussed in the previous section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the <em>dot product<\/em> and the <em>cross product<\/em>. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.<\/p>\n<p>The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Dot Product<\/h3>\n<p>The <strong>dot product<\/strong> of two vectors [latex]\\boldsymbol{v}=\\langle a,b\\rangle[\/latex] and [latex]\\boldsymbol{v}=\\langle c,d\\rangle[\/latex] is the sum of the product of the horizontal components and the product of the vertical components.<\/p>\n<p style=\"text-align: center;\">[latex]\\boldsymbol{v}\\cdot \\boldsymbol{u}=ac+bd[\/latex]<\/p>\n<p>To find the angle between the two vectors, use the formula below.<\/p>\n<p style=\"text-align: center;\">[latex]\\cos \\theta =\\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Finding the Dot Product of Two Vectors<\/h3>\n<p>Find the dot product of [latex]\\boldsymbol{v}=\\langle 5,12\\rangle[\/latex] and [latex]\\boldsymbol{u}=\\langle -3,4\\rangle[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q734442\">Show Solution<\/span><\/p>\n<div id=\"q734442\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the formula, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{v}\\cdot \\boldsymbol{u}&=\\langle 5,12\\rangle \\cdot \\langle -3,4\\rangle \\\\ &=5\\cdot \\left(-3\\right)+12\\cdot 4 \\\\ &=-15+48 \\\\ &=33 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Finding the Dot Product of Two Vectors and the Angle between Them<\/h3>\n<p>Find the dot product of <strong><em>v<\/em><\/strong><sub>1<\/sub> = 5<strong><em>i<\/em><\/strong> + 2<strong><em>j<\/em><\/strong> and <strong><em>v<\/em><\/strong><sub>2<\/sub> = 3<strong><em>i<\/em><\/strong> + 7<strong><em>j<\/em><\/strong>. Then, find the angle between the two vectors.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q817790\">Show Solution<\/span><\/p>\n<div id=\"q817790\" class=\"hidden-answer\" style=\"display: none\">\n<p>Finding the dot product, we multiply corresponding components.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{\\boldsymbol{v}}_{1}\\cdot {\\boldsymbol{v}}_{2}&=\\langle 5,2\\rangle \\cdot \\langle 3,7\\rangle \\\\ &=5\\cdot 3+2\\cdot 7 \\\\ &=15+14 \\\\ &=29 \\end{align}[\/latex]<\/p>\n<p>To find the angle between them, we use the formula [latex]\\cos \\theta =\\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\frac{\\boldsymbol{u}}{|\\boldsymbol{u}|}&=\\langle \\frac{5}{\\sqrt{29}}+\\frac{2}{\\sqrt{29}}\\rangle \\cdot \\langle \\frac{3}{\\sqrt{58}}+\\frac{7}{\\sqrt{58}}\\rangle \\\\ &=\\frac{5}{\\sqrt{29}}\\cdot \\frac{3}{\\sqrt{58}}+\\frac{2}{\\sqrt{29}}\\cdot \\frac{7}{\\sqrt{58}} \\\\ &=\\frac{15}{\\sqrt{1682}}+\\frac{14}{\\sqrt{1682}}=\\frac{29}{\\sqrt{1682}} \\\\ &=0.707107 \\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\theta={\\cos }^{-1}\\left(0.707107\\right)=45^\\circ[\/latex]<\/p>\n<div class=\"mceTemp\"><\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181206\/CNX_Precalc_Figure_08_08_0142.jpg\" alt=\"Plot showing the two position vectors (3,7) and (5,2) and the 45 degree angle between them.\" width=\"487\" height=\"403\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Finding the Angle between Two Vectors<\/h3>\n<p>Find the angle between [latex]\\boldsymbol{u}=\\langle -3,4\\rangle[\/latex] and [latex]\\boldsymbol{v}=\\langle 5,12\\rangle[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q953383\">Show Solution<\/span><\/p>\n<div id=\"q953383\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the formula, [latex]\\theta ={\\cos }^{-1}\\left(\\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}\\cdot \\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\right)[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} \\left(\\frac{\\boldsymbol{u}}{|\\boldsymbol{u}|}\\cdot \\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\right)&=\\frac{-3\\boldsymbol{i}+4\\boldsymbol{j}}{5}\\cdot \\frac{5\\boldsymbol{i}+12\\boldsymbol{j}}{13} \\\\ &=\\left(-\\frac{3}{5}\\cdot \\frac{5}{13}\\right)+\\left(\\frac{4}{5}\\cdot \\frac{12}{13}\\right) \\\\ &=-\\frac{15}{65}+\\frac{48}{65} \\\\ &=\\frac{33}{65}\\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\theta ={\\cos }^{-1}\\left(\\frac{33}{65}\\right) ={59.5}^{\\circ }[\/latex]<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" style=\"font-weight: bold; background-color: #f5f5f5; font-size: 0.9em; text-align: left;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181209\/CNX_Precalc_Figure_08_08_0132.jpg\" alt=\"Plot showing the two position vectors (-3,4) and (5,12) and the 59.5 degree angle between them.\" width=\"487\" height=\"628\" \/><\/p>\n<p><b>Figure 2<\/b><\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the dot product of [latex]\\boldsymbol{u}=[\/latex]\u00a04<strong><em>i<\/em><\/strong> &#8211; 3<strong><em>j<\/em><\/strong> and [latex]\\boldsymbol{v}=[\/latex]\u00a02<strong><em>i<\/em><\/strong> + 5<strong><em>j<\/em><\/strong>. Then, find the angle between the two vectors.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q453643\">Show Solution<\/span><\/p>\n<div id=\"q453643\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}=-7[\/latex]<\/p>\n<p>[latex]\\theta\\approx105^\\circ[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The dot product of two vectors is the product of the<em><strong> i <\/strong><\/em>terms plus the product of the<em><strong> j <\/strong><\/em>terms.<\/li>\n<li>We can use the dot product to find the angle between two vectors.<\/li>\n<li>Dot products are useful for many types of physics applications.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133447852\" class=\"definition\">\n<dt>dot product<\/dt>\n<dd id=\"fs-id1165133447857\">given two vectors, the sum of the product of the horizontal components and the product of the vertical components<\/dd>\n<dd><\/dd>\n<\/dl>\n<h2 style=\"text-align: center;\">Section 9.5 Homework Exercises<\/h2>\n<p>1.\u00a0Given [latex]\\boldsymbol{u}=\\boldsymbol{i}\u2212\\boldsymbol{j}[\/latex] and [latex]\\boldsymbol{v}=\\boldsymbol{i}+\\boldsymbol{j}[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.<\/p>\n<p>2.\u00a0Given [latex]\\boldsymbol{u}=3\\boldsymbol{i}\u22124\\boldsymbol{j}[\/latex] and [latex]\\boldsymbol{v}=\u22122\\boldsymbol{i}+3\\boldsymbol{j}[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.<\/p>\n<p>3.\u00a0Given [latex]\\boldsymbol{u}=\u2212\\boldsymbol{i}\u2212\\boldsymbol{j}[\/latex] and [latex]\\boldsymbol{v}=\\boldsymbol{i}+5\\boldsymbol{j}[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.<\/p>\n<p>4.\u00a0Given [latex]\\boldsymbol{u}=\\langle\u22122,4\\rangle[\/latex] and [latex]\\boldsymbol{v}=\\langle\u22123,1\\rangle[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.<\/p>\n<p>5.\u00a0Given [latex]\\boldsymbol{u}=\\langle\u22121,6\\rangle[\/latex] and [latex]\\boldsymbol{v}=\\langle 6,\u22121\\rangle[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex] and the angle between these two vectors.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"author":264444,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17982","chapter","type-chapter","status-publish","hentry"],"part":17975,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17982","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17982\/revisions"}],"predecessor-version":[{"id":18005,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17982\/revisions\/18005"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/17975"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17982\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=17982"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=17982"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=17982"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=17982"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}