Suppose I have two intersecting two dimensional planes in a four dimensional Euclidean space. It seems to me that there are two angles between these planes. If the two planes intersect in a line then one of those angles is zero. If the two angles are non-zero then the planes intersect in a point. If one plane is the $wx$ plane and the other the $yz$ plane then the two angles are both 90 degrees.
The best I can do is this. Each plane is defined by two orthonormal basis vectors. Project a basis vector onto the other plane. Then the angle is the $\arccos$ of the length of the projection. Do this with each basis vector to get the two angles. However I believe that the result depends on the choice of basis vectors.
What I need could be to find a unit vector in a plane so that it's projection onto the other plane is of maximum length. I don't know how to do that. I suppose find an expression for the length, take the derivative, and find where it is zero.
