{"id":2298,"date":"2025-08-12T21:36:31","date_gmt":"2025-08-12T21:36:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2298"},"modified":"2025-10-21T21:49:19","modified_gmt":"2025-10-21T21:49:19","slug":"operations-with-vectors-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/operations-with-vectors-learn-it-1\/","title":{"raw":"Operations with Vectors: Learn It 1","rendered":"Operations with Vectors: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Perform vector addition and scalar multiplication.<\/li>\r\n \t<li>Perform operations with vectors in terms of i and j .<\/li>\r\n \t<li>Find the dot product of two vectors.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Performing Vector Addition and Scalar Multiplication<\/h2>\r\nNow that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector [latex]\\boldsymbol{u}[\/latex] [latex]=\\langle x,y\\rangle [\/latex] as an arrow or directed line segment from the origin to the point [latex]\\left(x,y\\right)[\/latex], vectors can be situated anywhere in the plane. The sum of two vectors <strong><em>u<\/em><\/strong> and <strong><em>v<\/em><\/strong>, or <strong>vector addition<\/strong>, produces a third vector <strong><em>u<\/em><\/strong>+ <strong><em>v<\/em><\/strong>, the <strong>resultant<\/strong> vector.\r\n\r\nTo find <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, we first draw the vector <strong><em>u<\/em><\/strong>, and from the terminal end of <strong><em>u<\/em><\/strong>, we drawn the vector <strong><em>v<\/em><\/strong>. In other words, we have the initial point of <strong><em>v<\/em><\/strong> meet the terminal end of <strong><em>u<\/em><\/strong>. This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong> is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of <strong><em>u<\/em><\/strong> to the end of <strong><em>v<\/em><\/strong> in a straight path.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181145\/CNX_Precalc_Figure_08_08_0082.jpg\" alt=\"Diagrams of vector addition and subtraction. \" width=\"487\" height=\"149\" \/>\r\n\r\nVector subtraction is similar to vector addition. To find <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>, view it as <strong><em>u<\/em><\/strong> + (\u2212<strong><em>v<\/em><\/strong>). Adding \u2212<strong><em>v<\/em><\/strong> is reversing direction of <strong><em>v<\/em><\/strong> and adding it to the end of <strong><em>u<\/em><\/strong>. The new vector begins at the start of <strong><em>u<\/em><\/strong> and stops at the end point of \u2212<strong><em>v<\/em><\/strong>.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181147\/CNX_Precalc_Figure_08_08_0092.jpg\" alt=\"Showing vector addition and subtraction with parallelograms. For addition, the base is u, the side is v, the diagonal connecting the start of the base to the end of the side is u+v. For subtraction, thetop is u, the side is -v, and the diagonal connecting the start of the top to the end of the side is u-v.\" width=\"487\" height=\"128\" \/>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]\\boldsymbol{u}[\/latex] [latex]=\\langle 3,-2\\rangle [\/latex] and [latex]\\boldsymbol{v}[\/latex] [latex]=\\langle -1,4\\rangle [\/latex], find two new vectors <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, and <strong><em>u<\/em><\/strong> \u2212 <strong>v<\/strong>.[reveal-answer q=\"664138\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"664138\"]To find the sum of two vectors, we add the components. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{u}+\\boldsymbol{v}&amp;=\\langle 3,-2\\rangle +\\langle -1,4\\rangle \\\\ &amp;=\\langle 3+\\left(-1\\right),-2+4\\rangle \\\\ &amp;=\\langle 2,2\\rangle \\end{align}[\/latex].<\/p>\r\nTo find the difference of two vectors, add the negative components of [latex]v[\/latex] to [latex]u[\/latex]. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{u}+\\left(-\\boldsymbol{v}\\right)&amp;=\\langle 3,-2\\rangle +\\langle 1,-4\\rangle \\\\ &amp;=\\langle 3+1,-2+\\left(-4\\right)\\rangle \\\\ &amp;=\\langle 4,-6\\rangle \\end{align}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181150\/CNX_Precalc_Figure_08_08_0192.jpg\" alt=\"Further diagrams of vector addition and subtraction.\" width=\"731\" height=\"292\" \/> (a) Sum of two vectors (b) Difference of two vectors[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Performing Operations on Vectors in Terms of <em>i<\/em> and <em>j<\/em><\/h2>\r\nWhen vectors are written in terms of<em><strong> i <\/strong><\/em>and<strong><em> j<\/em><\/strong>, we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>adding and subtracting vectors in rectangular coordinates<\/h3>\r\nGiven <strong><em>v<\/em><\/strong> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> and <strong><em>u<\/em><\/strong> = <em>c<strong>i<\/strong><\/em> + <em>d<strong>j<\/strong><\/em>, then\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\boldsymbol{v}+\\boldsymbol{u}=\\left(a+c\\right)\\boldsymbol{i}+\\left(b+d\\right)\\boldsymbol{j}\\\\ \\boldsymbol{v}-\\boldsymbol{u}=\\left(a-c\\right)\\boldsymbol{i}+\\left(b-d\\right)\\boldsymbol{j}\\end{gathered}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the sum of [latex]{\\boldsymbol{v}}_{1}=2\\boldsymbol{i} - 3\\boldsymbol{j}[\/latex] and [latex]{\\boldsymbol{v}}_{2}=4\\boldsymbol{i}+5\\boldsymbol{j}[\/latex].[reveal-answer q=\"139436\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"139436\"]According to the formula, we have\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{\\boldsymbol{v}}_{1}+{\\boldsymbol{v}}_{2}&amp;=\\left(2+4\\right)\\boldsymbol{i}+\\left(-3+5\\right)\\boldsymbol{j} \\\\ &amp;=6\\boldsymbol{i}+2\\boldsymbol{j} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313888[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Perform vector addition and scalar multiplication.<\/li>\n<li>Perform operations with vectors in terms of i and j .<\/li>\n<li>Find the dot product of two vectors.<\/li>\n<\/ul>\n<\/section>\n<h2>Performing Vector Addition and Scalar Multiplication<\/h2>\n<p>Now that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector [latex]\\boldsymbol{u}[\/latex] [latex]=\\langle x,y\\rangle[\/latex] as an arrow or directed line segment from the origin to the point [latex]\\left(x,y\\right)[\/latex], vectors can be situated anywhere in the plane. The sum of two vectors <strong><em>u<\/em><\/strong> and <strong><em>v<\/em><\/strong>, or <strong>vector addition<\/strong>, produces a third vector <strong><em>u<\/em><\/strong>+ <strong><em>v<\/em><\/strong>, the <strong>resultant<\/strong> vector.<\/p>\n<p>To find <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, we first draw the vector <strong><em>u<\/em><\/strong>, and from the terminal end of <strong><em>u<\/em><\/strong>, we drawn the vector <strong><em>v<\/em><\/strong>. In other words, we have the initial point of <strong><em>v<\/em><\/strong> meet the terminal end of <strong><em>u<\/em><\/strong>. This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong> is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of <strong><em>u<\/em><\/strong> to the end of <strong><em>v<\/em><\/strong> in a straight path.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181145\/CNX_Precalc_Figure_08_08_0082.jpg\" alt=\"Diagrams of vector addition and subtraction.\" width=\"487\" height=\"149\" \/><\/p>\n<p>Vector subtraction is similar to vector addition. To find <strong><em>u<\/em><\/strong> \u2212 <strong><em>v<\/em><\/strong>, view it as <strong><em>u<\/em><\/strong> + (\u2212<strong><em>v<\/em><\/strong>). Adding \u2212<strong><em>v<\/em><\/strong> is reversing direction of <strong><em>v<\/em><\/strong> and adding it to the end of <strong><em>u<\/em><\/strong>. The new vector begins at the start of <strong><em>u<\/em><\/strong> and stops at the end point of \u2212<strong><em>v<\/em><\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181147\/CNX_Precalc_Figure_08_08_0092.jpg\" alt=\"Showing vector addition and subtraction with parallelograms. For addition, the base is u, the side is v, the diagonal connecting the start of the base to the end of the side is u+v. For subtraction, thetop is u, the side is -v, and the diagonal connecting the start of the top to the end of the side is u-v.\" width=\"487\" height=\"128\" \/><\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]\\boldsymbol{u}[\/latex] [latex]=\\langle 3,-2\\rangle[\/latex] and [latex]\\boldsymbol{v}[\/latex] [latex]=\\langle -1,4\\rangle[\/latex], find two new vectors <strong><em>u<\/em><\/strong> + <strong><em>v<\/em><\/strong>, and <strong><em>u<\/em><\/strong> \u2212 <strong>v<\/strong>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q664138\">Show Solution<\/button><\/p>\n<div id=\"q664138\" class=\"hidden-answer\" style=\"display: none\">To find the sum of two vectors, we add the components. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{u}+\\boldsymbol{v}&=\\langle 3,-2\\rangle +\\langle -1,4\\rangle \\\\ &=\\langle 3+\\left(-1\\right),-2+4\\rangle \\\\ &=\\langle 2,2\\rangle \\end{align}[\/latex].<\/p>\n<p>To find the difference of two vectors, add the negative components of [latex]v[\/latex] to [latex]u[\/latex]. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{u}+\\left(-\\boldsymbol{v}\\right)&=\\langle 3,-2\\rangle +\\langle 1,-4\\rangle \\\\ &=\\langle 3+1,-2+\\left(-4\\right)\\rangle \\\\ &=\\langle 4,-6\\rangle \\end{align}[\/latex]<\/p>\n<figure style=\"width: 731px\" 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\/27181150\/CNX_Precalc_Figure_08_08_0192.jpg\" alt=\"Further diagrams of vector addition and subtraction.\" width=\"731\" height=\"292\" \/><figcaption class=\"wp-caption-text\">(a) Sum of two vectors (b) Difference of two vectors<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<h2>Performing Operations on Vectors in Terms of <em>i<\/em> and <em>j<\/em><\/h2>\n<p>When vectors are written in terms of<em><strong> i <\/strong><\/em>and<strong><em> j<\/em><\/strong>, we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>adding and subtracting vectors in rectangular coordinates<\/h3>\n<p>Given <strong><em>v<\/em><\/strong> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> and <strong><em>u<\/em><\/strong> = <em>c<strong>i<\/strong><\/em> + <em>d<strong>j<\/strong><\/em>, then<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\boldsymbol{v}+\\boldsymbol{u}=\\left(a+c\\right)\\boldsymbol{i}+\\left(b+d\\right)\\boldsymbol{j}\\\\ \\boldsymbol{v}-\\boldsymbol{u}=\\left(a-c\\right)\\boldsymbol{i}+\\left(b-d\\right)\\boldsymbol{j}\\end{gathered}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the sum of [latex]{\\boldsymbol{v}}_{1}=2\\boldsymbol{i} - 3\\boldsymbol{j}[\/latex] and [latex]{\\boldsymbol{v}}_{2}=4\\boldsymbol{i}+5\\boldsymbol{j}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q139436\">Show Solution<\/button><\/p>\n<div id=\"q139436\" class=\"hidden-answer\" style=\"display: none\">According to the formula, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{\\boldsymbol{v}}_{1}+{\\boldsymbol{v}}_{2}&=\\left(2+4\\right)\\boldsymbol{i}+\\left(-3+5\\right)\\boldsymbol{j} \\\\ &=6\\boldsymbol{i}+2\\boldsymbol{j} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313888\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313888&theme=lumen&iframe_resize_id=ohm313888&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":520,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2298"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2298\/revisions"}],"predecessor-version":[{"id":4808,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2298\/revisions\/4808"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/520"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2298\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2298"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2298"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2298"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2298"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}