{"id":20232,"date":"2024-04-15T05:50:10","date_gmt":"2024-04-15T05:50:10","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20232"},"modified":"2024-04-15T05:50:10","modified_gmt":"2024-04-15T05:50:10","slug":"which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/","title":{"rendered":"Which of the following statements are correct regarding dot product of vectors? 1. Dot product is less than or equal to the product of magnitudes of two vectors. 2. When two vectors are perpendicular to each other, then their dot product is non-zero. 3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than \\[\\frac{\\pi }{2}\\]. 4. Dot product is equal to the product of one vector and the projection of the vector on the first one. Select the correct answer: A. 1, 2 and 3 only B. 1, 3 and 4 only C. 1, 2 and 4 only D. 2, 3 and 4 only"},"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_6a99a57426f79\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    1, 2 and 3 only                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    1, 3 and 4 only                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    1, 2 and 4 only                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"true\">\r\n                    2, 3 and 4 only                <\/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 $\\boxed{\\text{A}}$.<\/p>\n<p>The dot product of two vectors $\\mathbf{u}$ and $\\mathbf{v}$ is defined as $$\\mathbf{u} \\cdot \\mathbf{v} = \\|\\mathbf{u}\\| \\|\\mathbf{v}\\| \\cos \\theta,$$ where $\\|\\mathbf{u}\\|$ and $\\|\\mathbf{v}\\|$ are the magnitudes of $\\mathbf{u}$ and $\\mathbf{v}$, respectively, and $\\theta$ is the angle between $\\mathbf{u}$ and $\\mathbf{v}$.<\/p>\n<p>Statement 1 is correct. The dot product of two vectors is always less than or equal to the product of their magnitudes. This is because the cosine of the angle between two vectors is always less than or equal to 1, which is the case when the vectors are parallel.<\/p>\n<p>Statement 2 is incorrect. When two vectors are perpendicular to each other, then their dot product is zero. This is because the cosine of a right angle is 0.<\/p>\n<p>Statement 3 is correct. The dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than $\\frac{\\pi}{2}$. This is because the cosine of an angle is positive when the angle is less than $\\frac{\\pi}{2}$ and negative when the angle is greater than $\\frac{\\pi}{2}$.<\/p>\n<p>Statement 4 is incorrect. The dot product is not equal to the product of one vector and the projection of the vector <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> on the first one. The projection of a vector $\\mathbf{v}$ onto <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                    <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> a vector $\\mathbf{u}$ is given by $$\\mathbf{v}_\\text{proj} = \\frac{\\mathbf{u} \\cdot \\mathbf{v}}{\\|\\mathbf{u}\\|^2} \\mathbf{u}.$$ The dot product, on the other hand, is given by $$\\mathbf{u} \\cdot \\mathbf{v} = \\|\\mathbf{u}\\| \\|\\mathbf{v}\\| \\cos \\theta.$$ Therefore, the dot product and the projection of a vector onto another vector are not the same.<\/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>Which of the following statements are correct regarding dot product of vectors? 1. Dot product is less than or equal to the product of magnitudes of two vectors. 2. When two vectors are perpendicular to each other, then their dot product is non-zero. 3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than \\[\\frac{\\pi }{2}\\]. 4. Dot product is equal to the product of one vector and the projection of the vector on the first one. Select the correct answer: A. 1, 2 and 3 only B. 1, 3 and 4 only C. 1, 2 and 4 only D. 2, 3 and 4 only<\/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\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Which of the following statements are correct regarding dot product of vectors? 1. Dot product is less than or equal to the product of magnitudes of two vectors. 2. When two vectors are perpendicular to each other, then their dot product is non-zero. 3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than \\[\\frac{\\pi }{2}\\]. 4. Dot product is equal to the product of one vector and the projection of the vector on the first one. Select the correct answer: A. 1, 2 and 3 only B. 1, 3 and 4 only C. 1, 2 and 4 only D. 2, 3 and 4 only\" \/>\n<meta property=\"og:description\" content=\"Join Our Telegram Channel Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:50:10+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":"Which of the following statements are correct regarding dot product of vectors? 1. Dot product is less than or equal to the product of magnitudes of two vectors. 2. When two vectors are perpendicular to each other, then their dot product is non-zero. 3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than \\[\\frac{\\pi }{2}\\]. 4. Dot product is equal to the product of one vector and the projection of the vector on the first one. Select the correct answer: A. 1, 2 and 3 only B. 1, 3 and 4 only C. 1, 2 and 4 only D. 2, 3 and 4 only","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\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/","og_locale":"en_US","og_type":"article","og_title":"Which of the following statements are correct regarding dot product of vectors? 1. Dot product is less than or equal to the product of magnitudes of two vectors. 2. When two vectors are perpendicular to each other, then their dot product is non-zero. 3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than \\[\\frac{\\pi }{2}\\]. 4. Dot product is equal to the product of one vector and the projection of the vector on the first one. Select the correct answer: A. 1, 2 and 3 only B. 1, 3 and 4 only C. 1, 2 and 4 only D. 2, 3 and 4 only","og_description":"Join Our Telegram Channel Subscribe on YouTube","og_url":"https:\/\/exam.pscnotes.com\/mcq\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:50:10+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\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/","url":"https:\/\/exam.pscnotes.com\/mcq\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/","name":"Which of the following statements are correct regarding dot product of vectors? 1. Dot product is less than or equal to the product of magnitudes of two vectors. 2. When two vectors are perpendicular to each other, then their dot product is non-zero. 3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than \\[\\frac{\\pi }{2}\\]. 4. Dot product is equal to the product of one vector and the projection of the vector on the first one. Select the correct answer: A. 1, 2 and 3 only B. 1, 3 and 4 only C. 1, 2 and 4 only D. 2, 3 and 4 only","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:50:10+00:00","dateModified":"2024-04-15T05:50:10+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/which-of-the-following-statements-are-correct-regarding-dot-product-of-vectors-1-dot-product-is-less-than-or-equal-to-the-product-of-magnitudes-of-two-vectors-2-when-two-vectors-are-perpendicular\/#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":"Which of the following statements are correct regarding dot product of vectors? 1. Dot product is less than or equal to the product of magnitudes of two vectors. 2. When two vectors are perpendicular to each other, then their dot product is non-zero. 3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than \\[\\frac{\\pi }{2}\\]. 4. Dot product is equal to the product of one vector and the projection of the vector on the first one. Select the correct answer: A. 1, 2 and 3 only B. 1, 3 and 4 only C. 1, 2 and 4 only D. 2, 3 and 4 only"}]},{"@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\/20232"}],"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=20232"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20232\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=20232"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=20232"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=20232"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}