Question

In what ways can two planes intersect?

Answer 1

Typically two planes intersect along some line. If the planes happen to be parallel, then they will either not intersect at all or they will be the same plane.

Explanation:

If the equations of the two planes are given, and the left-hand sides of the equations are not multiples of each other (when written in standard form), then the planes will intersect along some line. The solution set of such a system (the intersection line) is typically represented in parametric form.

For example, if the equations are `x+2y+3z=4` and `x-4y+5z=2`, then you can subtract the second equation from the first to cancel the `x`'s and get `6y-2z=2`. You can then subtract this equation from 3 times the original first equation to eliminate the `2y` in the first equation, resulting in the system: `3x+11z=10`, `3y-z=1` (I also divided the previous equation by 2).

If you rewrite this system with `x` and `y` in terms of `z`, you'll get `x=10/3-11/3z`, `y=1/3 z+1/3`. In parametric form then, with `z` being the parameter (or "free variable"), the solution set is the line:

`(x,y,z)=(10/3-11/3z,1/3z+1/3,z)=(10/3,1/3,0)+z(-11/3,1/3,1)`.

The solution line therefore goes through the point `P=(10/3,1/3,0)` and is parallel to the vector `v =(-11/3,1/3,1)`.