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 Post subject: Plane equationPosted: Thu, 29 May 2003 21:06:27 UTC
Find the equation of the plane which goes through the line

x+y-z+1=0
2x-y+2z-1=0

and the midpoint of AB,where and

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 Post subject: Posted: Fri, 30 May 2003 01:45:29 UTC
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Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7949
Location: Lexington, MA
Hello, !

The planes x - y - z + 1 = 0 and 2x - y + 2z - 1 = 0 intersect in a line;
one set of parametric equations is: x = t, y = -1 - 4t, z = -3t.
The line has direction vector: L = <1,-4,3>.
One point on the line (when t = 0): P(0,-1,0)

The midpoint of A and B is: M(1,1,-1)
The vector PM = <1,2,-1>

The normal of the desired plane is normal to both L and PM.
The cross product of L and PM is: <10,-2,6> or <5,-1,3>.

The plane through (0,-1,0) with normal direction <5,-1,3> is:
5(x - 0) - 1(y + 1) + 3(z - 0) = 0
or:

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