{"id":195239,"date":"2025-02-28T05:24:14","date_gmt":"2025-02-28T05:24:14","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=195239"},"modified":"2025-02-28T05:24:17","modified_gmt":"2025-02-28T05:24:17","slug":"how-two-vectors-behave-if-their-dot-product-is-zero","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/28\/how-two-vectors-behave-if-their-dot-product-is-zero\/","title":{"rendered":"How two vectors behave if their dot product is zero"},"content":{"rendered":"\n

How two vectors behave if their dot product is zero<\/p>\n\n\n\n

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

If the dot product of two vectors is zero, it means that the vectors are orthogonal<\/strong> (perpendicular) to each other.<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

The dot product<\/strong> of two vectors A<\/strong> and B<\/strong> is given by:<\/p>\n\n\n\n

[
A \\cdot B = |A| |B| \\cos(\\theta)
]<\/p>\n\n\n\n

where:<\/p>\n\n\n\n

    \n
  • ( |A| ) and ( |B| ) are the magnitudes (lengths) of the vectors,<\/li>\n\n\n\n
  • ( \\theta ) is the angle between them,<\/li>\n\n\n\n
  • ( \\cos(\\theta) ) is the cosine of that angle.<\/li>\n<\/ul>\n\n\n\n

    For the dot product to be zero:<\/p>\n\n\n\n

    [
    |A| |B| \\cos(\\theta) = 0
    ]<\/p>\n\n\n\n

    Since the magnitudes of vectors cannot be zero unless they are zero vectors, the only way this equation holds is if:<\/p>\n\n\n\n

    [
    \\cos(\\theta) = 0
    ]<\/p>\n\n\n\n

    This occurs when:<\/p>\n\n\n\n

    [
    \\theta = 90^\\circ \\text{ (or ( \\frac{\\pi}{2} ) radians)}
    ]<\/p>\n\n\n\n

    Thus, if the dot product of two vectors is zero, the angle between them is 90\u00b0, meaning they are perpendicular (orthogonal) to each other.<\/p>\n\n\n\n

    Why Is This Important?<\/h3>\n\n\n\n
      \n
    1. In Physics:<\/strong> Perpendicular vectors appear in force calculations, work done by a force, and electromagnetic field components.<\/li>\n\n\n\n
    2. In Computer Graphics:<\/strong> Used in normal vector calculations to determine surface orientations.<\/li>\n\n\n\n
    3. In Linear Algebra:<\/strong> Orthogonal vectors play a key role in vector spaces and transformations.<\/li>\n<\/ol>\n\n\n\n

      Below is a visual representation of two perpendicular vectors, A<\/strong> and B<\/strong>, whose dot product is zero.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      The image above visually represents two perpendicular vectors A<\/strong> and B<\/strong>, originating from the same point (the origin), forming a 90-degree angle. This demonstrates that their dot product is zero, confirming their orthogonality. Let me know if you need any modifications!<\/p>\n","protected":false},"excerpt":{"rendered":"

      How two vectors behave if their dot product is zero The Correct Answer and Explanation is : If the dot product of two vectors is zero, it means that the vectors are orthogonal (perpendicular) to each other. Explanation: The dot product of two vectors A and B is given by: [A \\cdot B = |A| […]<\/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-195239","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195239","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=195239"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195239\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195239"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195239"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195239"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}