Point on a PlaneDate: 06/13/97 at 19:53:42 From: Brandon Schnell Subject: Finding a point on a plane How can I find a point on a plane? For example, if I know (x1,y1,z1), (x2,y2,z2) and (x3,y3,z3) define a plane, how could I solve for the unknown z of a point assuming I know the x and y coordinates? For example: (B)__________ (C) | / | * / | D/ | / | / | / | / | / | / |/ (A) A = (1,1,1) B = (1,10,10) C = (5,10,5) D = (2,9,x) Any help would be appreciated! Date: 06/14/97 at 08:15:40 From: Doctor Anthony Subject: Re: Finding a point on a plane You can use the three given points to derive the equation of the plane, then simply substitute the values x = 2, y = 9 into this equation to find z. Suppose the equation of the plane is: a(x-1) + b(y-1) + c(z-1) = 0 (1) Then (1,10,10) is on the plane: a(0) + b(9) + c(9) = 0 (2) Also (5,10,5) is on the plane: a(4) + b(9) + c(4) = 0 (3) Eliminating a,b,c from (1), (2) and (3) gives the determinant: | x-1 y-1 z-1 | | 0 9 9 | = 0 | 4 9 4 | So (x-1)(-45) - (y-1)(-36) + (z-1)(-36) = 0 Dividing through by -9: 5(x-1) - 4(y-1) + 4(z-1) = 0 5x - 4y + 4z - 5 = 0 (4) Now if x = 2, y = 9, equation (4) gives: 10 - 36 + 4z - 5 = 0 4z = 36 - 10 + 5 4z = 31 z = 31/4 Thus the coordinates of your third point are (2,9, 31/4). -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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