MA 242 - Distance in 3D How to find the distance between a...

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Distance in 3D How to find the distance between a point and a plane? Let P x p , y p , z p ± be any point in R 3 , and K : a x ? x 0 ± + b y ? y 0 ± + c z ? z 0 ± = 0 any plane. To find the distance between P and K , we first find the projection of P 0 P = ² x p ? x 0 , y p ? y 0 + z p ? z 0 ³ onto n = a , b , c the normal vector of K . Then d P , K ± = proj n P 0 P = n 6 P 0 P n . Example 1. Let P 1,2,3 ± and K : 2 x ? 2 ± + 1 y + 1 ± + 2 z ? 3 ± = 0 . Find the distance between P and K . Solution: Observe that P 0 2, ? 1,3 ± and n = 2,1,2 ,so P 0 P = 1 ? 2,2 ? ? 1 ± ,3 ? 3 = ? 1,3,0 . Therefore, d P , K ± = proj n P 0 P = n 6 P 0 P n = 2,1,2 6 ? 1,3,0 2 2 + 1 2 + 2 2 = 1 3 How to find the distance between a point and a line? Let P x p , y p , z p ± be any point in R 3 , and L : x = x 0 + at , y = y 0 + bt , and z = z 0 + ct any line. To find the distance between P and L , we first find the projection of P 0 P = ² x p ? x 0 , y p ? y 0 + z p ? z 0 ³ onto v = a , b , c the direction vector of L . Let w = proj v P 0 P = v 6 P 0 P v 2 v , then P 0 P and w
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This note was uploaded on 06/11/2010 for the course MA 242 taught by Professor Bliss during the Spring '08 term at N.C. State.

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MA 242 - Distance in 3D How to find the distance between a...

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