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Math Help - Equation of planes

  1. #1
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    Equation of planes

    Hi, can anyone explain to me why the equation for a plane3 which contains the intersection line of 2 other planes(plane1 and plane2), is the sum of the equations of plane 1 and plane 2?

    Note the planes all go through the origin.

    ie, if plane 1 is ax+by+cz=0 and if plane 2 is dx+ey+fz=0

    any plane containing the intersecting lines of plane 1 and plane 2 is:

    (a+d)x + (b+e)y + (c+f)z = 0

    Why?
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  2. #2
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    Re: Equation of planes

    Quote Originally Posted by darren86 View Post
    Hi, can anyone explain to me why the equation for a plane3 which contains the intersection line of 2 other planes(plane1 and plane2), is the sum of the equations of plane 1 and plane 2?

    Notation: \mathcal{R}=<x,y,z> and P is a point.

    Here are two planes N\cdot(\mathcal{R}-P)=0~\&~M\cdot(\mathcal{R}-P)=0,~N\not \parallel M, both containing P. The normals are N~\&~M

    The equation of the line of their intersection is P+t(N\times M).

    Because (N+M)\cdot(N\times M)=0 means the plane (N+M)\cdot(\mathcal{R}-P)=0 contains the line of intersection and has normal N+M.

    NOTE that answer is not unique. But it explains your question.
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