To find the implicit equation of a plane passing through the point (\u22125,\u22124,5)(-5, -4, 5)(\u22125,\u22124,5) and perpendicular to the line given by the parametric equation r(t)=(2\u22124t,2+t,4t\u22124)\\mathbf{r}(t) = (2 – 4t, 2 + t, 4t – 4)r(t)=(2\u22124t,2+t,4t\u22124), we can follow these steps:<\/p>\n\n\n\n
Step 1: Direction of the Line<\/h3>\n\n\n\n
The direction vector of the line r(t)=(2\u22124t,2+t,4t\u22124)\\mathbf{r}(t) = (2 – 4t, 2 + t, 4t – 4)r(t)=(2\u22124t,2+t,4t\u22124) can be extracted by taking the coefficients of ttt in each component of the parametric equation.<\/p>\n\n\n\n
- \n
- The xxx-component is 2\u22124t2 – 4t2\u22124t, so the coefficient of ttt is \u22124-4\u22124.<\/li>\n\n\n\n
- The yyy-component is 2+t2 + t2+t, so the coefficient of ttt is 111.<\/li>\n\n\n\n
- The zzz-component is 4t\u221244t – 44t\u22124, so the coefficient of ttt is 444.<\/li>\n<\/ul>\n\n\n\n
Thus, the direction vector of the line is d=(\u22124,1,4)\\mathbf{d} = (-4, 1, 4)d=(\u22124,1,4).<\/p>\n\n\n\n
Step 2: Normal Vector to the Plane<\/h3>\n\n\n\n
The plane is perpendicular to the line, which means the normal vector to the plane is parallel to the direction vector of the line. Therefore, the normal vector n\\mathbf{n}n of the plane is the same as the direction vector d\\mathbf{d}d, i.e., n=(\u22124,1,4)\\mathbf{n} = (-4, 1, 4)n=(\u22124,1,4).<\/p>\n\n\n\n
Step 3: Plane Equation<\/h3>\n\n\n\n
The general equation of a plane is given by:a(x\u2212x0)+b(y\u2212y0)+c(z\u2212z0)=0a(x – x_0) + b(y – y_0) + c(z – z_0) = 0a(x\u2212x0\u200b)+b(y\u2212y0\u200b)+c(z\u2212z0\u200b)=0<\/p>\n\n\n\n
where (x0,y0,z0)(x_0, y_0, z_0)(x0\u200b,y0\u200b,z0\u200b) is a point on the plane and (a,b,c)(a, b, c)(a,b,c) is the normal vector to the plane.<\/p>\n\n\n\n
- \n
- The normal vector is (\u22124,1,4)(-4, 1, 4)(\u22124,1,4), so a=\u22124a = -4a=\u22124, b=1b = 1b=1, and c=4c = 4c=4.<\/li>\n\n\n\n
- The point (\u22125,\u22124,5)(-5, -4, 5)(\u22125,\u22124,5) lies on the plane, so x0=\u22125x_0 = -5×0\u200b=\u22125, y0=\u22124y_0 = -4y0\u200b=\u22124, and z0=5z_0 = 5z0\u200b=5.<\/li>\n<\/ul>\n\n\n\n
Substitute these values into the equation of the plane:\u22124(x+5)+1(y+4)+4(z\u22125)=0-4(x + 5) + 1(y + 4) + 4(z – 5) = 0\u22124(x+5)+1(y+4)+4(z\u22125)=0<\/p>\n\n\n\n
Step 4: Simplify<\/h3>\n\n\n\n
Expand the terms:\u22124x\u221220+y+4+4z\u221220=0-4x – 20 + y + 4 + 4z – 20 = 0\u22124x\u221220+y+4+4z\u221220=0<\/p>\n\n\n\n
Combine like terms:\u22124x+y+4z\u221236=0-4x + y + 4z – 36 = 0\u22124x+y+4z\u221236=0<\/p>\n\n\n\n
Thus, the implicit equation of the plane is:\u22124x+y+4z=36-4x + y + 4z = 36\u22124x+y+4z=36<\/p>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n
The implicit equation of the plane passing through the point (\u22125,\u22124,5)(-5, -4, 5)(\u22125,\u22124,5) and perpendicular to the line r(t)=(2\u22124t,2+t,4t\u22124)\\mathbf{r}(t) = (2 – 4t, 2 + t, 4t – 4)r(t)=(2\u22124t,2+t,4t\u22124) is:\u22124x+y+4z=36-4x + y + 4z = 36\u22124x+y+4z=36<\/p>\n\n\n\n
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<\/figure>\n","protected":false},"excerpt":{"rendered":"Write an implicit equation for the plane passing through the point (-5, -4, 5) that is perpendicular to the line r(t) = (2 – 4t, 2 + t, 4t – 4). Plane equation: The Correct Answer and Explanation is: To find the implicit equation of a plane passing through the point (\u22125,\u22124,5)(-5, -4, 5)(\u22125,\u22124,5) 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-268221","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/268221","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=268221"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/268221\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=268221"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=268221"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=268221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}