{"id":20141,"date":"2024-04-15T05:48:56","date_gmt":"2024-04-15T05:48:56","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20141"},"modified":"2024-04-15T05:48:56","modified_gmt":"2024-04-15T05:48:56","slug":"let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/","title":{"rendered":"Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product. Then the determinant det \\[\\left[ {\\begin{array}{*{20}{c}} { < {\\text{x}},{\\text{x}} > }&#038;{ < {\\text{x}},{\\text{y}} > } \\\\ { < {\\text{y}},{\\text{x}} > }&#038;{ < {\\text{y}},{\\text{y}} > } \\end{array}} \\right].\\] A. is zero when x and y are linearly independent B. is positive when x and yare linearly independent C. is non-zero for all non-zero x and y D. is zero only when either x or y is zero"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a99476a241b8\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    is zero when x and y are linearly independent                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    is positive when x and yare linearly independent                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"true\">\r\n                    is non-zero for all non-zero x and y                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    is zero only when either x or y is zero                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is $\\boxed{\\text{C}}$.<\/p>\n<p>The determinant of a 2&#215;2 matrix can be computed using the formula:<\/p>\n<p>$$\\det \\begin{bmatrix} a &amp; b \\\\ c &amp; d \\end{bmatrix} = (a \\times d) &#8211; (b \\times c)$$<\/p>\n<p>In this case, we have:<\/p>\n<p>$$\\det \\begin{bmatrix} &lt;{\\text{x}},{\\text{x}} &gt; &amp; &lt;{\\text{x}},{\\text{y}} &gt; \\\\ &lt;{\\text{y}},{\\text{x}} &gt; &amp; &lt;{\\text{y}},{\\text{y}} &gt; \\end{bmatrix} <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n            <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div>         <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> = (&lt;{\\text{x}},{\\text{x}} &gt; \\times &lt;{\\text{y}},{\\text{y}} &gt;) &#8211; (&lt;{\\text{x}},{\\text{y}} &gt; \\times &lt;{\\text{y}},{\\text{x}} &gt;)$$<\/p>\n<p>We can use the definition of the dot product to rewrite this as:<\/p>\n<p>$$\\det \\begin{bmatrix} &lt;{\\text{x}},{\\text{x}} &gt; &amp; &lt;{\\text{x}},{\\text{y}} &gt; \\\\ &lt;{\\text{y}},{\\text{x}} &gt; &amp; &lt;{\\text{y}},{\\text{y}} &gt; \\end{bmatrix} = \\|{\\text{x}}\\|^2 \\|{\\text{y}}\\|^2 &#8211; &lt;{\\text{x}},{\\text{y}} &gt;^2$$<\/p>\n<p>The norm of a vector is its length, so $\\|{\\text{x}}\\|^2$ and $\\|{\\text{y}}\\|^2$ are always positive. The dot product of two vectors is zero if and only if the vectors are orthogonal (i.e., they are perpendicular). Therefore, the determinant is zero if and only if either $x$ or $y$ is zero, or if $x$ and $y$ are orthogonal.<\/p>\n<p>When $x$ and $y$ are linearly independent, they cannot be orthogonal, so the determinant cannot be zero. Therefore, the determinant is non-zero for all non-zero $x$ and $y$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Our Telegram Channel Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Let x and y be two vectors in a 3 dimensional space and  denote their dot product. Then the determinant det \\[\\left[ {\\begin{array}{*{20}{c}} { &lt; {\\text{x}},{\\text{x}} &gt; }&amp;{ &lt; {\\text{x}},{\\text{y}} &gt; } \\\\ { &lt; {\\text{y}},{\\text{x}} &gt; }&amp;{ &lt; {\\text{y}},{\\text{y}} &gt; } \\end{array}} \\right].\\] A. is zero when x and y are linearly independent B. is positive when x and yare linearly independent C. is non-zero for all non-zero x and y D. is zero only when either x or y is zero<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Let x and y be two vectors in a 3 dimensional space and  denote their dot product. Then the determinant det \\[\\left[ {\\begin{array}{*{20}{c}} { &lt; {\\text{x}},{\\text{x}} &gt; }&amp;{ &lt; {\\text{x}},{\\text{y}} &gt; } \\\\ { &lt; {\\text{y}},{\\text{x}} &gt; }&amp;{ &lt; {\\text{y}},{\\text{y}} &gt; } \\end{array}} \\right].\\] A. is zero when x and y are linearly independent B. is positive when x and yare linearly independent C. is non-zero for all non-zero x and y D. is zero only when either x or y is zero\" \/>\n<meta property=\"og:description\" content=\"Join Our Telegram Channel Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:48:56+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Let x and y be two vectors in a 3 dimensional space and  denote their dot product. Then the determinant det \\[\\left[ {\\begin{array}{*{20}{c}} { < {\\text{x}},{\\text{x}} > }&{ < {\\text{x}},{\\text{y}} > } \\\\ { < {\\text{y}},{\\text{x}} > }&{ < {\\text{y}},{\\text{y}} > } \\end{array}} \\right].\\] A. is zero when x and y are linearly independent B. is positive when x and yare linearly independent C. is non-zero for all non-zero x and y D. is zero only when either x or y is zero","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/","og_locale":"en_US","og_type":"article","og_title":"Let x and y be two vectors in a 3 dimensional space and  denote their dot product. Then the determinant det \\[\\left[ {\\begin{array}{*{20}{c}} { < {\\text{x}},{\\text{x}} > }&{ < {\\text{x}},{\\text{y}} > } \\\\ { < {\\text{y}},{\\text{x}} > }&{ < {\\text{y}},{\\text{y}} > } \\end{array}} \\right].\\] A. is zero when x and y are linearly independent B. is positive when x and yare linearly independent C. is non-zero for all non-zero x and y D. is zero only when either x or y is zero","og_description":"Join Our Telegram Channel Subscribe on YouTube","og_url":"https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:48:56+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/","url":"https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/","name":"Let x and y be two vectors in a 3 dimensional space and denote their dot product. Then the determinant det \\[\\left[ {\\begin{array}{*{20}{c}} { < {\\text{x}},{\\text{x}} > }&{ < {\\text{x}},{\\text{y}} > } \\\\ { < {\\text{y}},{\\text{x}} > }&{ < {\\text{y}},{\\text{y}} > } \\end{array}} \\right].\\] A. is zero when x and y are linearly independent B. is positive when x and yare linearly independent C. is non-zero for all non-zero x and y D. is zero only when either x or y is zero","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:48:56+00:00","dateModified":"2024-04-15T05:48:56+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/let-x-and-y-be-two-vectors-in-a-3-dimensional-space-and-denote-their-dot-product-then-the-determinant-det-left-beginarray20c-textxtextx-textxtext\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"mcq","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/"},{"@type":"ListItem","position":3,"name":"Engineering maths","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/"},{"@type":"ListItem","position":4,"name":"Calculus","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/calculus\/"},{"@type":"ListItem","position":5,"name":"Let x and y be two vectors in a 3 dimensional space and denote their dot product. Then the determinant det \\[\\left[ {\\begin{array}{*{20}{c}} { < {\\text{x}},{\\text{x}} > }&#038;{ < {\\text{x}},{\\text{y}} > } \\\\ { < {\\text{y}},{\\text{x}} > }&#038;{ < {\\text{y}},{\\text{y}} > } \\end{array}} \\right].\\] A. is zero when x and y are linearly independent B. is positive when x and yare linearly independent C. is non-zero for all non-zero x and y D. is zero only when either x or y is zero"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/d97f17072bfa490596c8f78363955d55?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/d97f17072bfa490596c8f78363955d55?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20141"}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=20141"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20141\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=20141"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=20141"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=20141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}