Is it possible for two non parallel planes to not intersect with each other in 3d?

Question

I read that two lines in 2d if they are not parallel they will always intersect but in 3d it is possible for two non parallel lines to not intersect but what about the planes , is it possible two non parallel planes in 3d to not intersect with each other?

Answer 1

No it is not possible and it can be shown by linear algebra since if you consider the following system of the two plane equations

• $ax+by+cz=d$
• $a'x+b'y+c'z=d'$

with a,b,c not multiple of a',b',c' the rank of the associated matrix is 2 and we always have infinitely many solutions.

Answer 2

It is not possible in three dimensional space, but it is possible in four and higher dimensional spaces. In four dimensional space, two planes need not be parallel to each other, and at the the same time they need not intersect each other. It is difficult to visualise this situation, but it can be proved.