As the title says, I have to prove that the cross product of normal vectors of two planes are parallel to the intersection of the two planes.
I used drawing on pen and paper, (which obviously is not a proof), and I still don't get how to prove this statement mathematically. Can anyone help me here? Thanks!
Best Answer
Conceptually, it is very easy. The normal vector to a plane is, by definition, orthogonal (perpendicular) to every line on the plane. The cross product of two vectors is, again by definition, orthogonal to the two vectors.
Therefore, the cross product of the two normal vectors will be parallel to each of the two planes. Which means it will also be parallel to the common line shared by the two planes - their line of intersection.
Proving it mathematically in your specific case is quite trivial, and you should be able to do it by following definitions. Just find the cross-product. Then show that the resulting vector is a scalar (non-zero constant number, which can be negative) multiple of the direction vector of the common line of the intersecting planes (which is given). That's it.
If you need more help, you will have to show more working. For one thing, actually work out the cross-product.