{"id":241012,"date":"2025-07-03T14:12:45","date_gmt":"2025-07-03T14:12:45","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=241012"},"modified":"2025-07-03T14:12:47","modified_gmt":"2025-07-03T14:12:47","slug":"find-the-projection-of-v-onto-u-and-find-the-angle-between-v-and-u","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/03\/find-the-projection-of-v-onto-u-and-find-the-angle-between-v-and-u\/","title":{"rendered":"find the projection of v onto u and find the angle between v and u."},"content":{"rendered":"\n<p>find the projection of v onto u and find the angle between v and u.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the <strong>projection<\/strong> of a vector v\\mathbf{v}v onto another vector u\\mathbf{u}u, we use the formula:projuv=v\u22c5uu\u22c5uu\\text{proj}_{\\mathbf{u}} \\mathbf{v} = \\frac{\\mathbf{v} \\cdot \\mathbf{u}}{\\mathbf{u} \\cdot \\mathbf{u}} \\mathbf{u}proju\u200bv=u\u22c5uv\u22c5u\u200bu<\/p>\n\n\n\n<p>Here\u2019s how to break it down:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Dot product v\u22c5u\\mathbf{v} \\cdot \\mathbf{u}v\u22c5u<\/strong>: This is the sum of the products of corresponding components of vectors v\\mathbf{v}v and u\\mathbf{u}u. If v=\u27e8v1,v2\u27e9\\mathbf{v} = \\langle v_1, v_2 \\ranglev=\u27e8v1\u200b,v2\u200b\u27e9 and u=\u27e8u1,u2\u27e9\\mathbf{u} = \\langle u_1, u_2 \\rangleu=\u27e8u1\u200b,u2\u200b\u27e9, the dot product is: v\u22c5u=v1u1+v2u2\\mathbf{v} \\cdot \\mathbf{u} = v_1 u_1 + v_2 u_2v\u22c5u=v1\u200bu1\u200b+v2\u200bu2\u200b<\/li>\n\n\n\n<li><strong>Magnitude squared of u\\mathbf{u}u<\/strong>: This is the sum of the squares of the components of u\\mathbf{u}u: u\u22c5u=u12+u22\\mathbf{u} \\cdot \\mathbf{u} = u_1^2 + u_2^2u\u22c5u=u12\u200b+u22\u200b<\/li>\n\n\n\n<li><strong>Projection formula<\/strong>: Using the values obtained from the above steps, we can find the projection of v\\mathbf{v}v onto u\\mathbf{u}u.<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Example:<\/h3>\n\n\n\n<p>Let\u2019s say v=\u27e83,4\u27e9\\mathbf{v} = \\langle 3, 4 \\ranglev=\u27e83,4\u27e9 and u=\u27e81,2\u27e9\\mathbf{u} = \\langle 1, 2 \\rangleu=\u27e81,2\u27e9.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Dot product v\u22c5u\\mathbf{v} \\cdot \\mathbf{u}v\u22c5u<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>v\u22c5u=3\u00d71+4\u00d72=3+8=11\\mathbf{v} \\cdot \\mathbf{u} = 3 \\times 1 + 4 \\times 2 = 3 + 8 = 11v\u22c5u=3\u00d71+4\u00d72=3+8=11<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li><strong>Magnitude squared of u\\mathbf{u}u<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>u\u22c5u=12+22=1+4=5\\mathbf{u} \\cdot \\mathbf{u} = 1^2 + 2^2 = 1 + 4 = 5u\u22c5u=12+22=1+4=5<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li><strong>Projection of v\\mathbf{v}v onto u\\mathbf{u}u<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>projuv=115\u27e81,2\u27e9=\u27e8115,225\u27e9\\text{proj}_{\\mathbf{u}} \\mathbf{v} = \\frac{11}{5} \\langle 1, 2 \\rangle = \\langle \\frac{11}{5}, \\frac{22}{5} \\rangleproju\u200bv=511\u200b\u27e81,2\u27e9=\u27e8511\u200b,522\u200b\u27e9<\/p>\n\n\n\n<p>So, the projection of v\\mathbf{v}v onto u\\mathbf{u}u is \u27e8115,225\u27e9\\langle \\frac{11}{5}, \\frac{22}{5} \\rangle\u27e8511\u200b,522\u200b\u27e9.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Angle Between v\\mathbf{v}v and u\\mathbf{u}u:<\/h3>\n\n\n\n<p>The formula for the angle \u03b8\\theta\u03b8 between two vectors v\\mathbf{v}v and u\\mathbf{u}u is:cos\u2061\u03b8=v\u22c5u\u2223v\u2223\u2223u\u2223\\cos \\theta = \\frac{\\mathbf{v} \\cdot \\mathbf{u}}{|\\mathbf{v}| |\\mathbf{u}|}cos\u03b8=\u2223v\u2223\u2223u\u2223v\u22c5u\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2223v\u2223|\\mathbf{v}|\u2223v\u2223 is the magnitude of v\\mathbf{v}v, calculated as v12+v22\\sqrt{v_1^2 + v_2^2}v12\u200b+v22\u200b\u200b<\/li>\n\n\n\n<li>\u2223u\u2223|\\mathbf{u}|\u2223u\u2223 is the magnitude of u\\mathbf{u}u, calculated as u12+u22\\sqrt{u_1^2 + u_2^2}u12\u200b+u22\u200b\u200b<\/li>\n<\/ul>\n\n\n\n<p>Using the values from the example:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Magnitude of v\\mathbf{v}v<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>\u2223v\u2223=32+42=9+16=25=5|\\mathbf{v}| = \\sqrt{3^2 + 4^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5\u2223v\u2223=32+42\u200b=9+16\u200b=25\u200b=5<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li><strong>Magnitude of u\\mathbf{u}u<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>\u2223u\u2223=12+22=1+4=5|\\mathbf{u}| = \\sqrt{1^2 + 2^2} = \\sqrt{1 + 4} = \\sqrt{5}\u2223u\u2223=12+22\u200b=1+4\u200b=5\u200b<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li><strong>Cosine of the angle<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>cos\u2061\u03b8=115\u00d75=115\u00d72.236\u22481111.18\u22480.983\\cos \\theta = \\frac{11}{5 \\times \\sqrt{5}} = \\frac{11}{5 \\times 2.236} \\approx \\frac{11}{11.18} \\approx 0.983cos\u03b8=5\u00d75\u200b11\u200b=5\u00d72.23611\u200b\u224811.1811\u200b\u22480.983<\/p>\n\n\n\n<ol start=\"4\" class=\"wp-block-list\">\n<li><strong>Angle \u03b8\\theta\u03b8<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>\u03b8=cos\u2061\u22121(0.983)\u224810.3\u2218\\theta = \\cos^{-1}(0.983) \\approx 10.3^\\circ\u03b8=cos\u22121(0.983)\u224810.3\u2218<\/p>\n\n\n\n<p>Thus, the angle between v\\mathbf{v}v and u\\mathbf{u}u is approximately 10.3\u221810.3^\\circ10.3\u2218.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner8-64.jpeg\" alt=\"\" class=\"wp-image-241018\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>find the projection of v onto u and find the angle between v and u. The Correct Answer and Explanation is: To find the projection of a vector v\\mathbf{v}v onto another vector u\\mathbf{u}u, we use the formula:projuv=v\u22c5uu\u22c5uu\\text{proj}_{\\mathbf{u}} \\mathbf{v} = \\frac{\\mathbf{v} \\cdot \\mathbf{u}}{\\mathbf{u} \\cdot \\mathbf{u}} \\mathbf{u}proju\u200bv=u\u22c5uv\u22c5u\u200bu Here\u2019s how to break it down: Example: Let\u2019s say v=\u27e83,4\u27e9\\mathbf{v} [&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-241012","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241012","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=241012"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241012\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=241012"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=241012"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=241012"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}