Math CentralQuandaries & Queries


Question from Sarim, a student:

How to find a intersection of two planes?


Suppose the two planes are

ax + by + cz = d and ex + fy + gz = h

then the vector (a, b, c) is normal to the first plane and (e, f, g) is normal to the second plane. Hence the vector

(p, q, r) = (a, b, c) times (e, f, g),

the cross product of (a, b, c) and (e, f, g), is in the direction of the line of intersection of the line of intersection of the planes. Thus the line of intersection is

x = x0 + p, y = y0 + q, z = z0 + r

where (x0, y0, z0) is a point on both planes. You can find a point (x0, y0, z0) in many ways. For example choose x = x0 to be any convenient number, substitute this value into the equations of the planes and then solve the resulting equations for y and z.


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