{"id":6903,"date":"2021-10-21T22:23:34","date_gmt":"2021-10-21T16:53:34","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6903"},"modified":"2021-10-25T10:07:07","modified_gmt":"2021-10-25T04:37:07","slug":"if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/","title":{"rendered":"If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|"},"content":{"rendered":"

Solution :<\/h2>\n

\\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\)<\/p>\n

\\(\\implies\\)\u00a0 \\(\\vec{c}\\perp\\vec{a}\\) , \\(\\vec{c}\\perp\\vec{b}\\) and \\(\\vec{a}\\perp\\vec{b}\\), \\(\\vec{a}\\perp\\vec{c}\\)<\/p>\n

\\(\\implies\\)\u00a0 \\(\\vec{a}\\perp\\vec{b}\\), \\(\\vec{b}\\perp\\vec{c}\\) and \\(\\vec{c}\\perp\\vec{a}\\)<\/p>\n

\\(\\implies\\)\u00a0 \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually perpendicular vectors.<\/p>\n

Again, \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\)<\/p>\n

\\(\\implies\\) |\\(\\vec{a}\\times\\vec{b}\\)| = |\\(\\vec{c}\\)| and |\\(\\vec{b}\\times\\vec{c}\\)| = |\\(\\vec{a}\\)|<\/p>\n

\\(\\implies\\)\u00a0 \\(|\\vec{a}||\\vec{b}|sin{\\pi\\over 2}\\) = |\\(\\vec{c}\\)| and \\(|\\vec{b}||\\vec{c}|sin{\\pi\\over 2}\\) = |\\(\\vec{a}\\)|\u00a0 (\\(\\because\\) \\(\\vec{a}\\perp\\vec{b}\\) and \\(\\vec{b}\\perp\\vec{c}\\))<\/p>\n

\\(\\implies\\)\u00a0 \\(|\\vec{a}||\\vec{b}|\\) = |\\(\\vec{c}\\)| and \\(|\\vec{b}||\\vec{c}|\\) = |\\(\\vec{a}\\)|<\/p>\n

\\(\\implies\\)\u00a0 \\({|\\vec{b}|}^2\\) |\\(\\vec{c}\\)| = |\\(\\vec{c}\\)|<\/p>\n

\\(\\implies\\)\u00a0 \\({|\\vec{b}|}^2\\) = 1<\/p>\n

\\(\\implies\\)\u00a0 \\(|\\vec{b}|\\) = 1<\/p>\n

putting in \\(|\\vec{a}||\\vec{b}|\\) = |\\(\\vec{c}\\)|<\/p>\n

\\(\\implies\\)\u00a0 \\(|\\vec{a}|\\) = |\\(\\vec{c}\\)|<\/p>\n

\n

Similar Questions<\/h3>\n

Find the angle between the vectors with the direction ratios proportional to 4, -3, 5 and 3, 4, 5.<\/a><\/p>\n

Find dot product of vectors \\(\\vec{a}\\) = \\(2\\hat{i}+2\\hat{j}-\\hat{k}\\) and \\(\\vec{b}\\) = \\(6\\hat{i}-3\\hat{j}+2\\hat{k}\\)<\/a><\/p>\n

For any three vectors \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) prove that [\\(\\vec{a}\\) + \\(\\vec{b}\\) \\(\\vec{b}\\) + \\(\\vec{c}\\) \\(\\vec{c}\\) + \\(\\vec{a}\\)] = 2[\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)]<\/a><\/p>\n

Find the vector equation of a line which passes through the point A (3, 4, -7) and B (1, -1, 6)<\/a><\/p>\n

Find the vector of magnitude 5 which are perpendicular to the vectors \\(\\vec{a}\\) = \\(2\\hat{i} + \\hat{j} \u2013 3\\hat{k}\\) and \\(\\vec{b}\\) = \\(\\hat{i} \u2013 2\\hat{j} + \\hat{k}\\)<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\) \\(\\implies\\)\u00a0 \\(\\vec{c}\\perp\\vec{a}\\) , \\(\\vec{c}\\perp\\vec{b}\\) and \\(\\vec{a}\\perp\\vec{b}\\), \\(\\vec{a}\\perp\\vec{c}\\) \\(\\implies\\)\u00a0 \\(\\vec{a}\\perp\\vec{b}\\), \\(\\vec{b}\\perp\\vec{c}\\) and \\(\\vec{c}\\perp\\vec{a}\\) \\(\\implies\\)\u00a0 \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually perpendicular vectors. Again, \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\) \\(\\implies\\) |\\(\\vec{a}\\times\\vec{b}\\)| = |\\(\\vec{c}\\)| and |\\(\\vec{b}\\times\\vec{c}\\)| = |\\(\\vec{a}\\)| \\(\\implies\\)\u00a0 \\(|\\vec{a}||\\vec{b}|sin{\\pi\\over 2}\\) = |\\(\\vec{c}\\)| and \\(|\\vec{b}||\\vec{c}|sin{\\pi\\over 2}\\) = |\\(\\vec{a}\\)|\u00a0 …<\/p>\n

If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,61],"tags":[],"yoast_head":"\nIf \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|<\/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:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|\" \/>\n<meta property=\"og:description\" content=\"Solution : (vec{a}timesvec{b}) = (vec{c}) and (vec{b}timesvec{c}) = (vec{a}) (implies)\u00a0 (vec{c}perpvec{a}) , (vec{c}perpvec{b}) and (vec{a}perpvec{b}), (vec{a}perpvec{c}) (implies)\u00a0 (vec{a}perpvec{b}), (vec{b}perpvec{c}) and (vec{c}perpvec{a}) (implies)\u00a0 (vec{a}), (vec{b}), (vec{c}) are mutually perpendicular vectors. Again, (vec{a}timesvec{b}) = (vec{c}) and (vec{b}timesvec{c}) = (vec{a}) (implies) |(vec{a}timesvec{b})| = |(vec{c})| and |(vec{b}timesvec{c})| = |(vec{a})| (implies)\u00a0 (|vec{a}||vec{b}|sin{piover 2}) = |(vec{c})| and (|vec{b}||vec{c}|sin{piover 2}) = |(vec{a})|\u00a0 … If (vec{a}), (vec{b}), (vec{c}) are three non zero vectors such that (vec{a}timesvec{b}) = (vec{c}) and (vec{b}timesvec{c}) = (vec{a}), prove that (vec{a}), (vec{b}), (vec{c}) are mutually at right angles and |(vec{b})| = 1 and |(vec{c})| = |(vec{a})| Read More »\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-21T16:53:34+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-25T04:37:07+00:00\" \/>\n<meta name=\"author\" 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\\\\(\\\\vec{c}\\\\) are three non zero vectors such that \\\\(\\\\vec{a}\\\\times\\\\vec{b}\\\\) = \\\\(\\\\vec{c}\\\\) and \\\\(\\\\vec{b}\\\\times\\\\vec{c}\\\\) = \\\\(\\\\vec{a}\\\\), prove that \\\\(\\\\vec{a}\\\\), \\\\(\\\\vec{b}\\\\), \\\\(\\\\vec{c}\\\\) are mutually at right angles and |\\\\(\\\\vec{b}\\\\)| = 1 and |\\\\(\\\\vec{c}\\\\)| = |\\\\(\\\\vec{a}\\\\)|\",\"datePublished\":\"2021-10-21T16:53:34+00:00\",\"dateModified\":\"2021-10-25T04:37:07+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/\"},\"wordCount\":335,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Maths Questions\",\"Vectors 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|\\\\(\\\\vec{c}\\\\)| = |\\\\(\\\\vec{a}\\\\)|\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-21T16:53:34+00:00\",\"dateModified\":\"2021-10-25T04:37:07+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"If \\\\(\\\\vec{a}\\\\), 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\/ Yoast SEO plugin. -->","yoast_head_json":{"title":"If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|","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:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/","og_locale":"en_US","og_type":"article","og_title":"If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|","og_description":"Solution : (vec{a}timesvec{b}) = (vec{c}) and (vec{b}timesvec{c}) = (vec{a}) (implies)\u00a0 (vec{c}perpvec{a}) , (vec{c}perpvec{b}) and (vec{a}perpvec{b}), (vec{a}perpvec{c}) (implies)\u00a0 (vec{a}perpvec{b}), (vec{b}perpvec{c}) and (vec{c}perpvec{a}) (implies)\u00a0 (vec{a}), (vec{b}), (vec{c}) are mutually perpendicular vectors. Again, (vec{a}timesvec{b}) = (vec{c}) and (vec{b}timesvec{c}) = (vec{a}) (implies) |(vec{a}timesvec{b})| = |(vec{c})| and |(vec{b}timesvec{c})| = |(vec{a})| (implies)\u00a0 (|vec{a}||vec{b}|sin{piover 2}) = |(vec{c})| and (|vec{b}||vec{c}|sin{piover 2}) = |(vec{a})|\u00a0 … If (vec{a}), (vec{b}), (vec{c}) are three non zero vectors such that (vec{a}timesvec{b}) = (vec{c}) and (vec{b}timesvec{c}) = (vec{a}), prove that (vec{a}), (vec{b}), (vec{c}) are mutually at right angles and |(vec{b})| = 1 and |(vec{c})| = |(vec{a})| Read More »","og_url":"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/","og_site_name":"Mathemerize","article_published_time":"2021-10-21T16:53:34+00:00","article_modified_time":"2021-10-25T04:37:07+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|","datePublished":"2021-10-21T16:53:34+00:00","dateModified":"2021-10-25T04:37:07+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/"},"wordCount":335,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"articleSection":["Maths Questions","Vectors Questions"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/","url":"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/","name":"If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-10-21T16:53:34+00:00","dateModified":"2021-10-25T04:37:07+00:00","breadcrumb":{"@id":"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/if-veca-vecb-vecc-are-three-non-zero-vectors-such-that-vecatimesvecb-vecc-and-vecbtimesvecc-veca-prove-that\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - 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