{"id":6390,"date":"2021-10-13T18:23:47","date_gmt":"2021-10-13T12:53:47","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6390"},"modified":"2022-01-16T17:05:29","modified_gmt":"2022-01-16T11:35:29","slug":"image-of-a-point-in-a-plane","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/image-of-a-point-in-a-plane\/","title":{"rendered":"Image of a Point in a Plane – Formula and Example"},"content":{"rendered":"

Here you will learn how to find image of a point in a plane formula with examples.<\/p>\n

Let’s begin –<\/p>\n

Image of a Point in a Plane<\/h2>\n

Let P and Q be two points and let \\(\\pi\\) be a plane such that<\/p>\n

(i)  line PQ is perpendicular to the plane \\(\\pi\\) and,<\/p>\n

(ii) mid-point of PQ lies on the plane \\(\\pi\\).<\/p>\n

Then, either of the point is the image of the other in the plane \\(\\pi\\).<\/p>\n

In order to find the image of a points \\((x_1, y_1, z_1)\\) in the plane ax + by + cz + d = 0, we may use the following algorithm.<\/p>\n

Algorithm : <\/strong><\/p>\n

\n

1). Write the equations of the line passing through P and normal to the given plane as \\(x – x_1\\over a\\) = \\(y – y_1\\over b\\) = \\(z – z_1\\over c\\).<\/p>\n

2). Write the coordinates of image Q as \\((x_1 + ar, y_1 + br, z_1 + cr)\\).<\/p>\n

3). Find the coordinates of the mid-point R of PQ.<\/p>\n

4). Obtain the value of r by substituting the coordinates of R in the equation of the plane.<\/p>\n

5). Put the value of r in the coordinates of Q.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Find the image of the point P (3, -2, 1) in the plane 3x – y + 4z = 2.<\/p>\n

Solution<\/span><\/strong> : Let Q be the image of P(3, -2, 1) in the plane 3x – y + 4z = 2. Then PQ is normal to the plane. Therefore, direction ratios of PQ are proportional to 3, -1, 4. Since PQ passes through P(3, -2, 1) and has direction ratios proportional to 3, -1, 4.<\/p>\n

Therefore, equation of PQ is<\/p>\n

\\(x – 3\\over 3\\) = \\(y + 2\\over -1\\) = \\(z – 1\\over 4\\) = r   (say)<\/p>\n

Let the coordinates of Q be (3r + 3, -r – 2, 4r + 1). Let R be the mid-point of PQ. Then, R lies on the plane 3x – y + 4z = 2. <\/p>\n

The coordinates of R are (\\({3r + 3 + 3\\over 3}, {-r – 2 – 2\\over 2}, {4r + 1 +1\\over 2}\\))<\/p>\n

or,  (\\({3r + 6\\over 3}, {-r – 4\\over 2}, {2r + 1}\\))<\/p>\n

Since r lies on the plane 3x – y + 4z = 2.<\/p>\n

\\(3({3r + 6\\over 2}) – ({-r – 4\\over 2})\\) + 4(2r + 1) = 2<\/p>\n

\\(\\implies\\) 13r = -13 \\(\\implies\\) r = – 1<\/p>\n

Putting r = -1 in (3r + 3, -r – 2, 4r + 1), we obtain the coordinates of Q as (0, – 1, -3).<\/p>\n

Hence, the image of (3, -2, 1) in the plane 3x –  y + 4z = 2 is (0, -1, -3).<\/p>\n\n\n

\n
Previous – Equation of Plane Containing Two Lines<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find image of a point in a plane formula with examples. Let’s begin – Image of a Point in a Plane Let P and Q be two points and let \\(\\pi\\) be a plane such that (i)  line PQ is perpendicular to the plane \\(\\pi\\) and, (ii) mid-point of …<\/p>\n

Image of a Point in a Plane – Formula and Example<\/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":[33],"tags":[],"yoast_head":"\nImage of a Point in a Plane - Formula and Example<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find image of a point in a plane formula with examples.\" \/>\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\/image-of-a-point-in-a-plane\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Image of a Point in a Plane - 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