{"id":20241,"date":"2024-04-15T05:50:17","date_gmt":"2024-04-15T05:50:17","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20241"},"modified":"2024-04-15T05:50:17","modified_gmt":"2024-04-15T05:50:17","slug":"the-area-of-a-triangle-formed-by-the-tips-of-vectors-overline-rma-rmoverline-rmb-and-overline-rmc-is-a-frac12left-overline-rma","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-area-of-a-triangle-formed-by-the-tips-of-vectors-overline-rma-rmoverline-rmb-and-overline-rmc-is-a-frac12left-overline-rma\/","title":{"rendered":"The area of a triangle formed by the tips of vectors \\[\\overline {\\rm{a}} {\\rm{,}}\\,\\overline {\\rm{b}} \\] and \\[\\overline {\\rm{c}} \\] is A. \\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{b}} } \\right) \\cdot \\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{c}} } \\right)\\] B. \\[\\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{c}} } \\right)} \\right|\\] C. \\[\\frac{1}{2}\\left| {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} \\times \\overline {\\rm{c}} } \\right|\\] D. \\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} } \\right) \\cdot \\overline {\\rm{c}} \\]"},"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_6a9928ab4b294\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    &#8221;[\frac{1}{2}left(                <\/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    &#8221; option2=&#8221;\\[\\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{c}} } \\right)} \\right|\\]&#8221; option3=&#8221;\\[\\frac{1}{2}\\left| {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} \\times \\overline {\\rm{c}} } \\right|\\]&#8221; option4=&#8221;\\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} } \\right) \\cdot \\overline {\\rm{c}} \\]&#8221; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The correct <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> answer is $\\boxed{\\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{c}} } \\right)} \\right|}$.<\/p>\n<p>The area of a triangle formed by the tips of vectors $\\overline {\\rm{a}} {\\rm{,}}\\,\\overline {\\rm{b}} ] and [\\overline {\\rm{c}} ] is given by the formula<\/p>\n<p>$$\\text{Area} = \\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{c}} } \\right)} \\right|$$<\/p>\n<p>where $\\times$ denotes the vector cross product.<\/p>\n<p>The vector cross product is a binary operation on two vectors that produces a vector perpendicular to both of the original vectors. The magnitude of the vector cross product is equal to the area of the parallelogram spanned by the two vectors, and the direction of the vector cross product is given by the right-hand rule.<\/p>\n<p>In this case, the vectors $\\overline {\\rm{a}} &#8211; \\overline {\\rm{b}} $ and $\\overline {\\rm{a}} &#8211; \\overline {\\rm{c}} $ are adjacent sides of the triangle, so the area of the triangle is given by the formula<\/p>\n<p>$$\\text{Area} = \\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} &#8211; <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{c}} } \\right)} \\right|$$<\/p>\n<p>The other options are incorrect. Option A is the formula for the scalar product of two vectors. Option B is the formula for the magnitude of the vector cross product. Option C is the formula for the volume of a parallelepiped spanned by three vectors.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#8221; option2=&#8221;\\[\\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} &#8211; \\overline {\\rm{c}} } \\right)} \\right|\\]&#8221; option3=&#8221;\\[\\frac{1}{2}\\left| {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} \\times Subscribe on YouTube \\overline {\\rm{c}} } \\right|\\]&#8221; option4=&#8221;\\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} } \\right) \\cdot \\overline {\\rm{c}} \\]&#8221; correct=&#8221;option1&#8243;]<\/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>The area of a triangle formed by the tips of vectors \\[\\overline {\\rm{a}} {\\rm{,}}\\,\\overline {\\rm{b}} \\] and \\[\\overline {\\rm{c}} \\] is A. \\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} - \\overline {\\rm{b}} } \\right) \\cdot \\left( {\\overline {\\rm{a}} - \\overline {\\rm{c}} } \\right)\\] B. \\[\\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} - \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} - \\overline {\\rm{c}} } \\right)} \\right|\\] C. \\[\\frac{1}{2}\\left| {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} \\times \\overline {\\rm{c}} } \\right|\\] D. \\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} } \\right) \\cdot \\overline {\\rm{c}} \\]<\/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\/the-area-of-a-triangle-formed-by-the-tips-of-vectors-overline-rma-rmoverline-rmb-and-overline-rmc-is-a-frac12left-overline-rma\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The area of a triangle formed by the tips of vectors \\[\\overline {\\rm{a}} {\\rm{,}}\\,\\overline {\\rm{b}} \\] and \\[\\overline {\\rm{c}} \\] is A. \\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} - \\overline {\\rm{b}} } \\right) \\cdot \\left( {\\overline {\\rm{a}} - \\overline {\\rm{c}} } \\right)\\] B. \\[\\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} - \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} - \\overline {\\rm{c}} } \\right)} \\right|\\] C. \\[\\frac{1}{2}\\left| {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} \\times \\overline {\\rm{c}} } \\right|\\] D. \\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} } \\right) \\cdot \\overline {\\rm{c}} \\]\" \/>\n<meta property=\"og:description\" content=\"&#8221; 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