Lines and Planes in 3-space

Finding the intersection of 2 planes

If you have two planes, they can either be parallel, co-incident, or intersect on a line. I'll ignore the first two, and concentrate on finding the equation of the line of intersection.

Consider the planes − 2 x + 3 y + z + 6 = 0 and 3 x − y + 2 z − 2 = 0. If this was 2 lines is 2-space we would just use substitution or elimination to find a point (this is what you would do to intersect 3 planes, btw). Unfortunately, we have more variables than equations.

We want our solution to look like $$(x,y,x) = [a_1,a_2,a_3] + t[d_1,d_2,d_3]$$, so we just pick one of our variables in the plane and set it to t, and then solve for x and y as functions of t. Lets choose z.

-2x + 3y + t + 6 = 0

3x - y + 2t - 2 = 0

we'll eliminate y by multiplying the second equation by 3 and addiong to the first

7x + 7t  = 0

x = -t

sub this into the second equation and solve for y

3(-t) - y + 2t - 2 = 0

y = -t - 2

This gives us our parametric equations, which are easy to turn into a vector equation

x = -t

y = -2 - t

z = t