{"id":20271,"date":"2024-04-15T05:50:41","date_gmt":"2024-04-15T05:50:41","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20271"},"modified":"2024-04-15T05:50:41","modified_gmt":"2024-04-15T05:50:41","slug":"if-overrightarrow-texta-and-overrightarrow-textb-are-two-arbitrary-vectors-with-magnitudes-a-and-b-respectively-left-overrightarrow-texta-times-overright","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/if-overrightarrow-texta-and-overrightarrow-textb-are-two-arbitrary-vectors-with-magnitudes-a-and-b-respectively-left-overrightarrow-texta-times-overright\/","title":{"rendered":"If $$\\overrightarrow {\\text{a}} $$ and $$\\overrightarrow {\\text{b}} $$ are two arbitrary vectors with magnitudes a and b, respectively, $${\\left| {\\overrightarrow {\\text{a}} \\times \\overrightarrow {\\text{b}} } \\right|^2}$$ will be equal to A. $${{\\text{a}}^2}{{\\text{b}}^2} – {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$ B. $${\\text{ab}} – \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$ C. $${{\\text{a}}^2}{{\\text{b}}^2} + {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$ D. $${\\text{ab}} + \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$"},"content":{"rendered":"

[amp_mcq option1=”$${{\\text{a}}^2}{{\\text{b}}^2} – {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$” option2=”$${\\text{ab}} – \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$” option3=”$${{\\text{a}}^2}{{\\text{b}}^2} + {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$” option4=”$${\\text{ab}} + \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$” correct=”option4″]<\/p>\n

The correct answer is $\\boxed{{\\text{a}}^2}{{\\text{b}}^2} – {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$.<\/p>\n

The cross product of two vectors $\\overrightarrow {\\text{a}}$ and $\\overrightarrow {\\text{b}}$ is denoted by $\\overrightarrow {\\text{a}} \\times \\overrightarrow {\\text{b}}$. The magnitude of the cross product is given by the formula<\/p>\n

$$\\left| {\\overrightarrow {\\text{a}} \\times \\overrightarrow {\\text{b}} } \\right| = \\sqrt{{\\text{a}}^2{{\\text{b}}^2} – {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}}$$<\/p>\n

Therefore, the square of the magnitude of the cross product is given by<\/p>\n

$${\\left| {\\overrightarrow {\\text{a}} \\times \\overrightarrow {\\text{b}} } \\right|^2} = {\\text{a}}^2{{\\text{b}}^2} – {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$<\/p>\n

Option A is incorrect because it does not include the factor of $-1$ in front of the dot product. Option B is incorrect because it does not include the square of the magnitudes of the vectors. Option C is incorrect because it does not include the negative sign in front of the dot product. Option D is incorrect because it includes the dot product instead of the cross product.<\/p>\n","protected":false},"excerpt":{"rendered":"

[amp_mcq option1=”$${{\\text{a}}^2}{{\\text{b}}^2} – {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$” option2=”$${\\text{ab}} – \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$” option3=”$${{\\text{a}}^2}{{\\text{b}}^2} + {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$” option4=”$${\\text{ab}} + \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$” correct=”option4″]<\/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":[],"class_list":["post-20271","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"\nIf $$\\overrightarrow {\\text{a}} $$ and $$\\overrightarrow {\\text{b}} $$ are two arbitrary vectors with magnitudes a and b, respectively, $${\\left| {\\overrightarrow {\\text{a}} \\times \\overrightarrow {\\text{b}} } \\right|^2}$$ will be equal to A. $${{\\text{a}}^2}{{\\text{b}}^2} - {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$ B. $${\\text{ab}} - \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$ C. $${{\\text{a}}^2}{{\\text{b}}^2} + {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$ D. $${\\text{ab}} + \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$<\/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\/if-overrightarrow-texta-and-overrightarrow-textb-are-two-arbitrary-vectors-with-magnitudes-a-and-b-respectively-left-overrightarrow-texta-times-overright\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If $$\\overrightarrow {\\text{a}} $$ and $$\\overrightarrow {\\text{b}} $$ are two arbitrary vectors with magnitudes a and b, respectively, $${\\left| {\\overrightarrow {\\text{a}} \\times \\overrightarrow {\\text{b}} } \\right|^2}$$ will be equal to A. $${{\\text{a}}^2}{{\\text{b}}^2} - {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$ B. $${\\text{ab}} - \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$ C. $${{\\text{a}}^2}{{\\text{b}}^2} + {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$ D. $${\\text{ab}} + \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=”$${{text{a}}^2}{{text{b}}^2} – {left( {overrightarrow {text{a}} cdot overrightarrow {text{b}} } right)^2}$$” option2=”$${text{ab}} – overrightarrow {text{a}} cdot overrightarrow {text{b}} $$” option3=”$${{text{a}}^2}{{text{b}}^2} + {left( {overrightarrow {text{a}} cdot overrightarrow {text{b}} } right)^2}$$” option4=”$${text{ab}} + overrightarrow {text{a}} cdot overrightarrow {text{b}} $$” correct=”option4″]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/if-overrightarrow-texta-and-overrightarrow-textb-are-two-arbitrary-vectors-with-magnitudes-a-and-b-respectively-left-overrightarrow-texta-times-overright\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:50:41+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":"If $$\\overrightarrow {\\text{a}} $$ and $$\\overrightarrow {\\text{b}} $$ are two arbitrary vectors with magnitudes a and b, respectively, $${\\left| {\\overrightarrow {\\text{a}} \\times \\overrightarrow {\\text{b}} } \\right|^2}$$ will be equal to A. $${{\\text{a}}^2}{{\\text{b}}^2} - {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$ B. $${\\text{ab}} - \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$ C. $${{\\text{a}}^2}{{\\text{b}}^2} + {\\left( {\\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} } \\right)^2}$$ D. $${\\text{ab}} + \\overrightarrow {\\text{a}} \\cdot \\overrightarrow {\\text{b}} $$","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\/if-overrightarrow-texta-and-overrightarrow-textb-are-two-arbitrary-vectors-with-magnitudes-a-and-b-respectively-left-overrightarrow-texta-times-overright\/","og_locale":"en_US","og_type":"article","og_title":"If $$\\overrightarrow {\\text{a}} $$ and $$\\overrightarrow {\\text{b}} $$ are two arbitrary vectors with magnitudes a and b, respectively, $${\\left| {\\overrightarrow {\\text{a}} \\times \\overrightarrow {\\text{b}} } \\right|^2}$$ will be equal to A. $${{\\text{a}}^2}{{\\text{b}}^2} - 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