1. ## 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?

2. ## Re: Equation of planes

Originally Posted by darren86
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: $\displaystyle \mathcal{R}=<x,y,z>$ and $\displaystyle P$ is a point.

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

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

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