{"id":189726,"date":"2025-02-10T07:54:23","date_gmt":"2025-02-10T07:54:23","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=189726"},"modified":"2025-02-10T07:54:25","modified_gmt":"2025-02-10T07:54:25","slug":"find-the-projection-of-u-onto-v-and-then-find-the-vector-component-of-u-orthogonal-to-v","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/10\/find-the-projection-of-u-onto-v-and-then-find-the-vector-component-of-u-orthogonal-to-v\/","title":{"rendered":"\u00a0Find the projection of u onto v and then find the vector component of u orthogonal to v"},"content":{"rendered":"\n<p>\u00a0Find the projection of u onto v and then find the vector component of u orthogonal to v. Show the steps of your calculation. Sketch a figure and label u, v, the projection of u onto v and the vector component of u orthogonal to v. u=(2,3) and v=(5,1) proj, (u) = w = vector component of u orthogonal to V=<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To solve for the projection of vector <strong>u<\/strong> onto vector <strong>v<\/strong>, and the vector component of <strong>u<\/strong> orthogonal to <strong>v<\/strong>, we will follow these steps:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Formula for the projection of <strong>u<\/strong> onto <strong>v<\/strong><\/h3>\n\n\n\n<p>The formula for the projection of vector <strong>u<\/strong> onto vector <strong>v<\/strong> is given by:<\/p>\n\n\n\n<p>[<br>\\text{proj}_{v}(u) = \\frac{u \\cdot v}{v \\cdot v} v<br>]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( u \\cdot v ) is the dot product of <strong>u<\/strong> and <strong>v<\/strong><\/li>\n\n\n\n<li>( v \\cdot v ) is the dot product of <strong>v<\/strong> with itself, i.e., the squared magnitude of <strong>v<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Calculate the dot products<\/h3>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>u<\/strong> = (2, 3)<\/li>\n\n\n\n<li><strong>v<\/strong> = (5, 1)<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Dot product ( u \\cdot v ):<\/h4>\n\n\n\n<p>[<br>u \\cdot v = (2)(5) + (3)(1) = 10 + 3 = 13<br>]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Dot product ( v \\cdot v ):<\/h4>\n\n\n\n<p>[<br>v \\cdot v = (5)(5) + (1)(1) = 25 + 1 = 26<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Calculate the projection of <strong>u<\/strong> onto <strong>v<\/strong><\/h3>\n\n\n\n<p>Using the formula:<\/p>\n\n\n\n<p>[<br>\\text{proj}_{v}(u) = \\frac{13}{26} v = \\frac{1}{2} (5, 1) = \\left( \\frac{5}{2}, \\frac{1}{2} \\right)<br>]<\/p>\n\n\n\n<p>So, the projection of <strong>u<\/strong> onto <strong>v<\/strong> is ( \\left( \\frac{5}{2}, \\frac{1}{2} \\right) ).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Calculate the component of <strong>u<\/strong> orthogonal to <strong>v<\/strong><\/h3>\n\n\n\n<p>The component of <strong>u<\/strong> orthogonal to <strong>v<\/strong> can be found by subtracting the projection of <strong>u<\/strong> onto <strong>v<\/strong> from <strong>u<\/strong>:<\/p>\n\n\n\n<p>[<br>w = u &#8211; \\text{proj}_{v}(u)<br>]<\/p>\n\n\n\n<p>[<br>w = (2, 3) &#8211; \\left( \\frac{5}{2}, \\frac{1}{2} \\right) = \\left( 2 &#8211; \\frac{5}{2}, 3 &#8211; \\frac{1}{2} \\right)<br>]<\/p>\n\n\n\n<p>[<br>w = \\left( \\frac{4}{2} &#8211; \\frac{5}{2}, \\frac{6}{2} &#8211; \\frac{1}{2} \\right) = \\left( -\\frac{1}{2}, \\frac{5}{2} \\right)<br>]<\/p>\n\n\n\n<p>So, the vector component of <strong>u<\/strong> orthogonal to <strong>v<\/strong> is ( \\left( -\\frac{1}{2}, \\frac{5}{2} \\right) ).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 5: Conclusion<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The projection of <strong>u<\/strong> onto <strong>v<\/strong> is ( \\left( \\frac{5}{2}, \\frac{1}{2} \\right) ).<\/li>\n\n\n\n<li>The vector component of <strong>u<\/strong> orthogonal to <strong>v<\/strong> is ( \\left( -\\frac{1}{2}, \\frac{5}{2} \\right) ).<\/li>\n<\/ul>\n\n\n\n<p>Now, let&#8217;s create a sketch to visualize this:<\/p>\n\n\n\n<p>I&#8217;ll generate a plot with the vectors <strong>u<\/strong>, <strong>v<\/strong>, the projection of <strong>u<\/strong> onto <strong>v<\/strong>, and the orthogonal component.<\/p>\n\n\n\n<p>In the plot above, you can see the following:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The red vector <strong>u<\/strong> = (2, 3).<\/li>\n\n\n\n<li>The blue vector <strong>v<\/strong> = (5, 1).<\/li>\n\n\n\n<li>The green vector represents the projection of <strong>u<\/strong> onto <strong>v<\/strong> (( \\text{proj}_v(u) = \\left( \\frac{5}{2}, \\frac{1}{2} \\right) )).<\/li>\n\n\n\n<li>The purple vector represents the component of <strong>u<\/strong> orthogonal to <strong>v<\/strong> (( w = \\left( -\\frac{1}{2}, \\frac{5}{2} \\right) )).<\/li>\n<\/ul>\n\n\n\n<p>This visualization helps to understand how <strong>u<\/strong> breaks down into two parts: one parallel to <strong>v<\/strong> (the projection) and one orthogonal to <strong>v<\/strong> (the orthogonal component).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00a0Find the projection of u onto v and then find the vector component of u orthogonal to v. Show the steps of your calculation. Sketch a figure and label u, v, the projection of u onto v and the vector component of u orthogonal to v. u=(2,3) and v=(5,1) proj, (u) = w = vector [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-189726","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189726","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=189726"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189726\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=189726"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=189726"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=189726"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}