# How do you find parametric equations for the line of intersection of two planes 2x - 2y + z = 1, and 2x + y - 3z = 3?

##### 1 Answer

#### Explanation:

for the line, we will need

A) a point that it actually passes through, say

B) a vector describing the direction in which it travels, say

....such that the line itself is

For **completely arbitrary** point, so here we choose the point at which z = 0

This means that the plane equations, namely

become

and these we solve as simultaneous equations to get

so

for

In the drawing below, we are looking right down the line of intersection, and we get an idea as to why the cross product of the normals of the red and blue planes generates a third vector, perpendicular to the normal vectors, that defines the direction of the line of intersection.

for a generalised plane

so for

and for

the cross product

so we combine this all as indicated in

There are an infinite number of ways of expressing this line.

The important bit, I guess is the method.