In what ways can two planes intersect?

1 Answer
Jan 25, 2016

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)#.