{"id":253421,"date":"2025-07-12T05:04:23","date_gmt":"2025-07-12T05:04:23","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=253421"},"modified":"2025-07-12T05:04:26","modified_gmt":"2025-07-12T05:04:26","slug":"separate-the-vectors-with-a-comma-and-do-not-pull-any-numbers-or-roots-outside-of-the-vector","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/12\/separate-the-vectors-with-a-comma-and-do-not-pull-any-numbers-or-roots-outside-of-the-vector\/","title":{"rendered":"Separate the vectors with a comma and do not pull any numbers or roots outside of the vector."},"content":{"rendered":"\n

Find two unit vectors orthogonal to both u=<4,-3,1> and v=<0,1,8> . Give exact values (no decimals). Separate the vectors with a comma and do not pull any numbers or roots outside of the vector.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To find two unit vectors orthogonal to both vectors u = <4, -3, 1><\/strong> and v = <0, 1, 8><\/strong>, we can follow these steps:<\/p>\n\n\n\n

Step 1: Find the Cross Product<\/h3>\n\n\n\n

The cross product of two vectors u<\/strong> and v<\/strong> gives a vector that is orthogonal to both. The formula for the cross product of two vectors u = <u\u2081, u\u2082, u\u2083><\/strong> and v = <v\u2081, v\u2082, v\u2083><\/strong> is: u\u00d7v=\u2223i^j^k^u1u2u3v1v2v3\u2223u \\times v = \\left| \\begin{matrix} \\hat{i} & \\hat{j} & \\hat{k} \\\\ u\u2081 & u\u2082 & u\u2083 \\\\ v\u2081 & v\u2082 & v\u2083 \\end{matrix} \\right|u\u00d7v=\u200bi^u1\u200bv1\u200b\u200bj^\u200bu2\u200bv2\u200b\u200bk^u3\u200bv3\u200b\u200b\u200b<\/p>\n\n\n\n

Now, applying the formula for the vectors u = <4, -3, 1><\/strong> and v = <0, 1, 8><\/strong>: u\u00d7v=i^\u2223\u22123118\u2223\u2212j^\u22234108\u2223+k^\u22234\u2212301\u2223u \\times v = \\hat{i} \\left| \\begin{matrix} -3 & 1 \\\\ 1 & 8 \\end{matrix} \\right| – \\hat{j} \\left| \\begin{matrix} 4 & 1 \\\\ 0 & 8 \\end{matrix} \\right| + \\hat{k} \\left| \\begin{matrix} 4 & -3 \\\\ 0 & 1 \\end{matrix} \\right|u\u00d7v=i^\u200b\u221231\u200b18\u200b\u200b\u2212j^\u200b\u200b40\u200b18\u200b\u200b+k^\u200b40\u200b\u221231\u200b\u200b<\/p>\n\n\n\n

Calculating each of the 2×2 determinants: i^:(\u22123)(8)\u2212(1)(1)=\u221224\u22121=\u221225\\hat{i}: (-3)(8) – (1)(1) = -24 – 1 = -25i^:(\u22123)(8)\u2212(1)(1)=\u221224\u22121=\u221225 j^:(4)(8)\u2212(0)(1)=32\u22120=32\\hat{j}: (4)(8) – (0)(1) = 32 – 0 = 32j^\u200b:(4)(8)\u2212(0)(1)=32\u22120=32 k^:(4)(1)\u2212(\u22123)(0)=4\u22120=4\\hat{k}: (4)(1) – (-3)(0) = 4 – 0 = 4k^:(4)(1)\u2212(\u22123)(0)=4\u22120=4<\/p>\n\n\n\n

Thus, the cross product is: u\u00d7v=\u221225i^\u221232j^+4k^=<\u221225,\u221232,4>u \\times v = -25\\hat{i} – 32\\hat{j} + 4\\hat{k} = <-25, -32, 4>u\u00d7v=\u221225i^\u221232j^\u200b+4k^=<\u221225,\u221232,4><\/p>\n\n\n\n

Step 2: Find the Magnitude of the Cross Product<\/h3>\n\n\n\n

Next, we calculate the magnitude (or length) of the vector u \u00d7 v<\/strong>. The magnitude of a vector <x, y, z><\/strong> is given by: \u2225u\u00d7v\u2225=x2+y2+z2\\| u \\times v \\| = \\sqrt{x^2 + y^2 + z^2}\u2225u\u00d7v\u2225=x2+y2+z2\u200b<\/p>\n\n\n\n

For <-25, -32, 4><\/strong>, we have: \u2225u\u00d7v\u2225=(\u221225)2+(\u221232)2+42=625+1024+16=1665\\| u \\times v \\| = \\sqrt{(-25)^2 + (-32)^2 + 4^2} = \\sqrt{625 + 1024 + 16} = \\sqrt{1665}\u2225u\u00d7v\u2225=(\u221225)2+(\u221232)2+42\u200b=625+1024+16\u200b=1665\u200b<\/p>\n\n\n\n

Step 3: Normalize the Cross Product<\/h3>\n\n\n\n

To find a unit vector, we divide the vector u \u00d7 v<\/strong> by its magnitude. The unit vector is: u^=11665\u27e8\u221225,\u221232,4\u27e9=\u27e8\u2212251665,\u2212321665,41665\u27e9\\hat{u} = \\frac{1}{\\sqrt{1665}} \\langle -25, -32, 4 \\rangle = \\left\\langle \\frac{-25}{\\sqrt{1665}}, \\frac{-32}{\\sqrt{1665}}, \\frac{4}{\\sqrt{1665}} \\right\\rangleu^=1665\u200b1\u200b\u27e8\u221225,\u221232,4\u27e9=\u27e81665\u200b\u221225\u200b,1665\u200b\u221232\u200b,1665\u200b4\u200b\u27e9<\/p>\n\n\n\n

Step 4: Find the Second Unit Vector<\/h3>\n\n\n\n

The second unit vector orthogonal to both u<\/strong> and v<\/strong> is simply the negative of the first. Thus, it is: \u2212u^=\u27e8251665,321665,\u221241665\u27e9-\\hat{u} = \\left\\langle \\frac{25}{\\sqrt{1665}}, \\frac{32}{\\sqrt{1665}}, \\frac{-4}{\\sqrt{1665}} \\right\\rangle\u2212u^=\u27e81665\u200b25\u200b,1665\u200b32\u200b,1665\u200b\u22124\u200b\u27e9<\/p>\n\n\n\n

Final Answer:<\/h3>\n\n\n\n

The two unit vectors orthogonal to both u<\/strong> and v<\/strong> are: \u27e8\u2212251665,\u2212321665,41665\u27e9,\u27e8251665,321665,\u221241665\u27e9\\left\\langle \\frac{-25}{\\sqrt{1665}}, \\frac{-32}{\\sqrt{1665}}, \\frac{4}{\\sqrt{1665}} \\right\\rangle, \\left\\langle \\frac{25}{\\sqrt{1665}}, \\frac{32}{\\sqrt{1665}}, \\frac{-4}{\\sqrt{1665}} \\right\\rangle\u27e81665\u200b\u221225\u200b,1665\u200b\u221232\u200b,1665\u200b4\u200b\u27e9,\u27e81665\u200b25\u200b,1665\u200b32\u200b,1665\u200b\u22124\u200b\u27e9<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Find two unit vectors orthogonal to both u=<4,-3,1> and v=<0,1,8> . Give exact values (no decimals). Separate the vectors with a comma and do not pull any numbers or roots outside of the vector. The Correct Answer and Explanation is: To find two unit vectors orthogonal to both vectors u = <4, -3, 1> and […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-253421","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/253421","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=253421"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/253421\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=253421"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=253421"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=253421"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}