{"id":241021,"date":"2025-07-03T14:14:44","date_gmt":"2025-07-03T14:14:44","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=241021"},"modified":"2025-07-03T14:14:46","modified_gmt":"2025-07-03T14:14:46","slug":"vector-projection-definition","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/03\/vector-projection-definition\/","title":{"rendered":"Vector Projection Definition"},"content":{"rendered":"\n

Vector Projection Definition: For two vectors U and V, the projection of V onto U is a vector that captures how much of V points in the direction of U. It is notated as proj(U)(V). The projection can be calculated as: proj(U)(V) = (U dot V \/ ||U||^2) * U For each set of vectors below, compute proj(U)(V) and plot U, V, and proj(U)(V) on the same plot: (a) U = [J], V = J<\/a> U = [B], V = 4<\/a> U = [D], V = []<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Vector Projection Overview<\/h3>\n\n\n\n

The projection<\/strong> of vector V<\/strong> onto vector U<\/strong> is the vector that represents the component of V<\/strong> in the direction of U<\/strong>. This is mathematically expressed as:proj(U)(V)=(U\u22c5V\u2225U\u22252)\u22c5U\\text{proj}(U)(V) = \\left( \\frac{U \\cdot V}{\\|U\\|^2} \\right) \\cdot Uproj(U)(V)=(\u2225U\u22252U\u22c5V\u200b)\u22c5U<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • U\u22c5VU \\cdot VU\u22c5V is the dot product<\/strong> of the vectors U<\/strong> and V<\/strong>.<\/li>\n\n\n\n
  • \u2225U\u2225\\|U\\|\u2225U\u2225 is the magnitude<\/strong> of the vector U<\/strong>, and \u2225U\u22252\\|U\\|^2\u2225U\u22252 is its squared magnitude.<\/li>\n\n\n\n
  • The result is a vector parallel to U<\/strong> representing how much of V<\/strong> is in the same direction as U<\/strong>.<\/li>\n<\/ul>\n\n\n\n

    Part (a) U=[J],V=[J]U = [J], V = [J]U=[J],V=[J]<\/h3>\n\n\n\n
      \n
    • Here, both vectors U<\/strong> and V<\/strong> are the same, so the projection of V<\/strong> onto U<\/strong> is simply V<\/strong>. The dot product U\u22c5VU \\cdot VU\u22c5V will be the square of the magnitude of U<\/strong> (or V<\/strong>) since they are the same vector.<\/li>\n<\/ul>\n\n\n\n

      Given U=[J]U = [J]U=[J] and V=[J]V = [J]V=[J]:<\/p>\n\n\n\n

        \n
      • Dot product U\u22c5V=J\u22c5J=J2U \\cdot V = J \\cdot J = J^2U\u22c5V=J\u22c5J=J2<\/li>\n\n\n\n
      • Magnitude of U<\/strong>: \u2225U\u2225=J\\|U\\| = J\u2225U\u2225=J<\/li>\n\n\n\n
      • Projection: proj(U)(V)=(J2J2)\u22c5[J]=[J]\\text{proj}(U)(V) = \\left( \\frac{J^2}{J^2} \\right) \\cdot [J] = [J]proj(U)(V)=(J2J2\u200b)\u22c5[J]=[J]<\/li>\n<\/ul>\n\n\n\n

        Thus, the projection of V<\/strong> onto U<\/strong> is [J]<\/strong>, the same as V<\/strong>.<\/p>\n\n\n\n

        Part (b) U=[B],V=[4]U = [B], V = [4]U=[B],V=[4]<\/h3>\n\n\n\n

        Given U = [B]<\/strong> and V = [4]<\/strong>:<\/p>\n\n\n\n

          \n
        • Dot product U\u22c5V=B\u22c54=4BU \\cdot V = B \\cdot 4 = 4BU\u22c5V=B\u22c54=4B<\/li>\n\n\n\n
        • Magnitude of U<\/strong>: \u2225U\u2225=B\\|U\\| = B\u2225U\u2225=B<\/li>\n\n\n\n
        • Projection: proj(U)(V)=(4BB2)\u22c5[B]=(4B)\u22c5[B]\\text{proj}(U)(V) = \\left( \\frac{4B}{B^2} \\right) \\cdot [B] = \\left( \\frac{4}{B} \\right) \\cdot [B]proj(U)(V)=(B24B\u200b)\u22c5[B]=(B4\u200b)\u22c5[B]<\/li>\n<\/ul>\n\n\n\n

          So, the projection of V<\/strong> onto U<\/strong> is a vector that is parallel to U<\/strong>, with magnitude proportional to 4B\\frac{4}{B}B4\u200b.<\/p>\n\n\n\n

          Part (c) U=[D],V=[]U = [D], V = []U=[D],V=[]<\/h3>\n\n\n\n

          For this case, since V<\/strong> is the zero vector (indicated by []<\/strong>), the projection of any vector V<\/strong> onto any vector U<\/strong> is the zero vector.<\/p>\n\n\n\n

            \n
          • The dot product U\u22c5V=D\u22c50=0U \\cdot V = D \\cdot 0 = 0U\u22c5V=D\u22c50=0<\/li>\n\n\n\n
          • The projection will be: proj(U)(V)=(0D2)\u22c5[D]=[0]\\text{proj}(U)(V) = \\left( \\frac{0}{D^2} \\right) \\cdot [D] = [0]proj(U)(V)=(D20\u200b)\u22c5[D]=[0]<\/li>\n<\/ul>\n\n\n\n

            Thus, the projection is the zero vector, [0]<\/strong>.<\/p>\n\n\n\n

            Visualization<\/h3>\n\n\n\n

            You can plot the vectors U<\/strong>, V<\/strong>, and proj(U)(V)<\/strong> on a graph as follows:<\/p>\n\n\n\n

              \n
            • For part (a), U<\/strong> and V<\/strong> are identical, so their vectors are exactly the same on the plot.<\/li>\n\n\n\n
            • For part (b), the projection proj(U)(V)<\/strong> is a scalar multiple of U<\/strong>, and can be plotted as a vector that points in the same direction as U<\/strong> but with a different magnitude.<\/li>\n\n\n\n
            • For part (c), V<\/strong> is a zero vector, and the projection will also be a zero vector, meaning no visible projection on the plot.<\/li>\n<\/ul>\n\n\n\n

              The general rule of thumb is that the projection vector proj(U)(V)<\/strong> always lies along the direction of U<\/strong>, regardless of V<\/strong>‘s direction.<\/p>\n\n\n\n

              \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

              Vector Projection Definition: For two vectors U and V, the projection of V onto U is a vector that captures how much of V points in the direction of U. It is notated as proj(U)(V). The projection can be calculated as: proj(U)(V) = (U dot V \/ ||U||^2) * U For each set of vectors […]<\/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-241021","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241021","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=241021"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241021\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=241021"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=241021"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=241021"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}