{"id":268225,"date":"2025-07-24T01:58:08","date_gmt":"2025-07-24T01:58:08","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=268225"},"modified":"2025-07-24T01:58:10","modified_gmt":"2025-07-24T01:58:10","slug":"an-implicit-equation-for-the-plane-passing-through-the-points-5-2-1-4-4-2-and-4-2-3-is-4x1y2z0","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/24\/an-implicit-equation-for-the-plane-passing-through-the-points-5-2-1-4-4-2-and-4-2-3-is-4x1y2z0\/","title":{"rendered":"An implicit equation for the plane passing through the points (5, 2, -1), (4, 4, 2), and (4, 2, -3) is -4x+1y+2z=0"},"content":{"rendered":"\n

An implicit equation for the plane passing through the points (5, 2, -1), (4, 4, 2), and (4, 2, -3) is -4x+1y+2z=0<\/p>\n\n\n\n

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

ChatGPT said:<\/h6>\n\n\n\n

To find the implicit equation of the plane passing through the points (5,2,\u22121)(5, 2, -1)(5,2,\u22121), (4,4,2)(4, 4, 2)(4,4,2), and (4,2,\u22123)(4, 2, -3)(4,2,\u22123), we can follow these steps:<\/p>\n\n\n\n

Step 1: Find two vectors that lie on the plane<\/h3>\n\n\n\n

We start by creating two vectors from the given points. Let’s use A=(5,2,\u22121)\\mathbf{A} = (5, 2, -1)A=(5,2,\u22121), B=(4,4,2)\\mathbf{B} = (4, 4, 2)B=(4,4,2), and C=(4,2,\u22123)\\mathbf{C} = (4, 2, -3)C=(4,2,\u22123).<\/p>\n\n\n\n

    \n
  1. Vector AB\\mathbf{AB}AB<\/strong> is the vector from AAA to BBB: AB=B\u2212A=(4\u22125,4\u22122,2\u2212(\u22121))=(\u22121,2,3)\\mathbf{AB} = B – A = (4 – 5, 4 – 2, 2 – (-1)) = (-1, 2, 3)AB=B\u2212A=(4\u22125,4\u22122,2\u2212(\u22121))=(\u22121,2,3)<\/li>\n\n\n\n
  2. Vector AC\\mathbf{AC}AC<\/strong> is the vector from AAA to CCC: AC=C\u2212A=(4\u22125,2\u22122,\u22123\u2212(\u22121))=(\u22121,0,\u22122)\\mathbf{AC} = C – A = (4 – 5, 2 – 2, -3 – (-1)) = (-1, 0, -2)AC=C\u2212A=(4\u22125,2\u22122,\u22123\u2212(\u22121))=(\u22121,0,\u22122)<\/li>\n<\/ol>\n\n\n\n

    Step 2: Find the normal vector to the plane<\/h3>\n\n\n\n

    To find the normal vector n\\mathbf{n}n to the plane, we take the cross product of vectors AB\\mathbf{AB}AB and AC\\mathbf{AC}AC:n=AB\u00d7AC\\mathbf{n} = \\mathbf{AB} \\times \\mathbf{AC}n=AB\u00d7AC<\/p>\n\n\n\n

    We compute the cross product:n=\u2223ijk\u2212123\u221210\u22122\u2223\\mathbf{n} = \\begin{vmatrix} \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\ -1 & 2 & 3 \\\\ -1 & 0 & -2 \\end{vmatrix}n=\u200bi\u22121\u22121\u200bj20\u200bk3\u22122\u200b\u200b<\/p>\n\n\n\n

    Expanding the determinant:n=i\u2223230\u22122\u2223\u2212j\u2223\u221213\u22121\u22122\u2223+k\u2223\u221212\u221210\u2223\\mathbf{n} = \\mathbf{i} \\begin{vmatrix} 2 & 3 \\\\ 0 & -2 \\end{vmatrix} – \\mathbf{j} \\begin{vmatrix} -1 & 3 \\\\ -1 & -2 \\end{vmatrix} + \\mathbf{k} \\begin{vmatrix} -1 & 2 \\\\ -1 & 0 \\end{vmatrix}n=i\u200b20\u200b3\u22122\u200b\u200b\u2212j\u200b\u22121\u22121\u200b3\u22122\u200b\u200b+k\u200b\u22121\u22121\u200b20\u200b\u200bn=i(2(\u22122)\u22123(0))\u2212j(\u22121(\u22122)\u22123(\u22121))+k(\u22121(0)\u22122(\u22121))\\mathbf{n} = \\mathbf{i} (2(-2) – 3(0)) – \\mathbf{j} (-1(-2) – 3(-1)) + \\mathbf{k} (-1(0) – 2(-1))n=i(2(\u22122)\u22123(0))\u2212j(\u22121(\u22122)\u22123(\u22121))+k(\u22121(0)\u22122(\u22121))n=i(\u22124)\u2212j(2+3)+k(0+2)\\mathbf{n} = \\mathbf{i}(-4) – \\mathbf{j}(2 + 3) + \\mathbf{k}(0 + 2)n=i(\u22124)\u2212j(2+3)+k(0+2)n=\u22124i\u22125j+2k\\mathbf{n} = -4\\mathbf{i} – 5\\mathbf{j} + 2\\mathbf{k}n=\u22124i\u22125j+2k<\/p>\n\n\n\n

    Thus, the normal vector is n=(\u22124,\u22125,2)\\mathbf{n} = (-4, -5, 2)n=(\u22124,\u22125,2).<\/p>\n\n\n\n

    Step 3: Write the equation of the plane<\/h3>\n\n\n\n

    The equation of the plane is given by the dot product of the normal vector and the vector from any point on the plane to a general point (x,y,z)(x, y, z)(x,y,z):n\u22c5(x\u2212x1,y\u2212y1,z\u2212z1)=0\\mathbf{n} \\cdot (x – x_1, y – y_1, z – z_1) = 0n\u22c5(x\u2212x1\u200b,y\u2212y1\u200b,z\u2212z1\u200b)=0<\/p>\n\n\n\n

    Using point A(5,2,\u22121)A(5, 2, -1)A(5,2,\u22121), we get:\u22124(x\u22125)\u22125(y\u22122)+2(z+1)=0-4(x – 5) – 5(y – 2) + 2(z + 1) = 0\u22124(x\u22125)\u22125(y\u22122)+2(z+1)=0<\/p>\n\n\n\n

    Expanding:\u22124x+20\u22125y+10+2z+2=0-4x + 20 – 5y + 10 + 2z + 2 = 0\u22124x+20\u22125y+10+2z+2=0\u22124x\u22125y+2z+32=0-4x – 5y + 2z + 32 = 0\u22124x\u22125y+2z+32=0<\/p>\n\n\n\n

    So the implicit equation of the plane is:\u22124x\u22125y+2z=\u221232-4x – 5y + 2z = -32\u22124x\u22125y+2z=\u221232<\/p>\n\n\n\n

    or equivalently:4x+5y\u22122z=324x + 5y – 2z = 324x+5y\u22122z=32<\/p>\n\n\n\n

    Conclusion:<\/h3>\n\n\n\n

    The equation \u22124x+y+2z=0-4x + y + 2z = 0\u22124x+y+2z=0 does not match the correct result from the steps above. The correct equation of the plane passing through the points (5,2,\u22121)(5, 2, -1)(5,2,\u22121), (4,4,2)(4, 4, 2)(4,4,2), and (4,2,\u22123)(4, 2, -3)(4,2,\u22123) is:4x+5y\u22122z=324x + 5y – 2z = 324x+5y\u22122z=32<\/p>\n\n\n\n

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

    An implicit equation for the plane passing through the points (5, 2, -1), (4, 4, 2), and (4, 2, -3) is -4x+1y+2z=0 The Correct Answer and Explanation is: ChatGPT said: To find the implicit equation of the plane passing through the points (5,2,\u22121)(5, 2, -1)(5,2,\u22121), (4,4,2)(4, 4, 2)(4,4,2), and (4,2,\u22123)(4, 2, -3)(4,2,\u22123), we can follow […]<\/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-268225","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/268225","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=268225"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/268225\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=268225"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=268225"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=268225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}