{"id":190532,"date":"2025-02-12T11:17:36","date_gmt":"2025-02-12T11:17:36","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=190532"},"modified":"2025-02-12T11:17:39","modified_gmt":"2025-02-12T11:17:39","slug":"find-the-projection-of-v-onto-u","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/12\/find-the-projection-of-v-onto-u\/","title":{"rendered":"Find the projection of v onto u"},"content":{"rendered":"\n<p>Find the projection of v onto u.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-409.png\" alt=\"\" class=\"wp-image-190533\"\/><\/figure>\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>The projection of vector <strong>v<\/strong> onto vector <strong>u<\/strong> is a vector that represents the component of <strong>v<\/strong> in the direction of <strong>u<\/strong>. This projection is calculated using the formula:<\/p>\n\n\n\n<p>[ \\text{proj}_{\\mathbf{u}} \\mathbf{v} = \\frac{\\mathbf{v} \\cdot \\mathbf{u}}{\\mathbf{u} \\cdot \\mathbf{u}} \\mathbf{u} ]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>v<\/strong> \u00b7 <strong>u<\/strong> is the dot product of <strong>v<\/strong> and <strong>u<\/strong>.<\/li>\n\n\n\n<li><strong>u<\/strong> \u00b7 <strong>u<\/strong> is the dot product of <strong>u<\/strong> with itself, which equals the square of the magnitude of <strong>u<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p><strong>Step-by-Step Calculation:<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>**Compute the Dot Product of *<em>v<\/em>* and <strong>u<\/strong>:**<br>[ \\mathbf{v} \\cdot \\mathbf{u} = v_1 \\times u_1 + v_2 \\times u_2 + v_3 \\times u_3 ]<br>This operation multiplies corresponding components of <strong>v<\/strong> and <strong>u<\/strong> and sums the results.<\/li>\n\n\n\n<li>**Compute the Dot Product of *<em>u<\/em>* with Itself:**<br>[ \\mathbf{u} \\cdot \\mathbf{u} = u_1^2 + u_2^2 + u_3^2 ]<br>This gives the square of the magnitude of <strong>u<\/strong>.<\/li>\n\n\n\n<li><strong>Calculate the Scalar Projection:<\/strong><br>[ \\text{Scalar Projection} = \\frac{\\mathbf{v} \\cdot \\mathbf{u}}{\\mathbf{u} \\cdot \\mathbf{u}} ]<br>This scalar represents the magnitude of the projection of <strong>v<\/strong> onto <strong>u<\/strong>.<\/li>\n\n\n\n<li><strong>Find the Vector Projection:<\/strong><br>[ \\text{proj}_{\\mathbf{u}} \\mathbf{v} = \\text{Scalar Projection} \\times \\mathbf{u} ]<br>This step scales <strong>u<\/strong> by the scalar projection to obtain the vector projection.<\/li>\n<\/ol>\n\n\n\n<p><strong>Geometric Interpretation:<\/strong><\/p>\n\n\n\n<p>The vector projection of <strong>v<\/strong> onto <strong>u<\/strong> represents the component of <strong>v<\/strong> that points in the same direction as <strong>u<\/strong>. Geometrically, it is the shadow or footprint of <strong>v<\/strong> when a light source is placed directly above <strong>u<\/strong>. This projection is particularly useful in physics and engineering for resolving forces, velocities, or other vector quantities into components along specific directions.<\/p>\n\n\n\n<p><strong>Example:<\/strong><\/p>\n\n\n\n<p>Consider vectors <strong>v<\/strong> = (3, 4, 0) and <strong>u<\/strong> = (1, 2, 0).<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>**Dot Product of *<em>v<\/em>* and <strong>u<\/strong>:**<br>[ \\mathbf{v} \\cdot \\mathbf{u} = (3 \\times 1) + (4 \\times 2) + (0 \\times 0) = 3 + 8 = 11 ]<\/li>\n\n\n\n<li>**Dot Product of *<em>u<\/em>* with Itself:**<br>[ \\mathbf{u} \\cdot \\mathbf{u} = (1^2) + (2^2) + (0^2) = 1 + 4 = 5 ]<\/li>\n\n\n\n<li><strong>Scalar Projection:<\/strong><br>[ \\text{Scalar Projection} = \\frac{11}{5} = 2.2 ]<\/li>\n\n\n\n<li><strong>Vector Projection:<\/strong><br>[ \\text{proj}_{\\mathbf{u}} \\mathbf{v} = 2.2 \\times (1, 2, 0) = (2.2, 4.4, 0) ]<\/li>\n<\/ol>\n\n\n\n<p>Thus, the projection of <strong>v<\/strong> onto <strong>u<\/strong> is the vector (2.2, 4.4, 0).<\/p>\n\n\n\n<p>For a visual explanation and further clarification, you might find the following video helpful:<\/p>\n\n\n\n<p>\ue200video\ue202How to Find the Projection of u Onto v and the Vector Component of u Orthogonal to v\ue202turn0search4\ue201<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find the projection of v onto u. The Correct Answer and Explanation is : The projection of vector v onto vector u is a vector that represents the component of v in the direction of u. This projection is calculated using the formula: [ \\text{proj}_{\\mathbf{u}} \\mathbf{v} = \\frac{\\mathbf{v} \\cdot \\mathbf{u}}{\\mathbf{u} \\cdot \\mathbf{u}} \\mathbf{u} ] Where: [&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-190532","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190532","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=190532"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190532\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=190532"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=190532"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=190532"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}