If A and B are parallel and equal, then A ∩ B = A = B. If A and B are parallel but not equal, then A ∩ B = Ø.
If A and B are not parallel and not equal, then they will intersect. This intersection will be a line.
How do you figure out if the planes are parallel? They are parallel if the spaces orthogonal to them are the same, i.e. ⊥A = ⊥B. Note that ⊥A and ⊥B are lines. What if you form a plane from the span of those two lines? That would be the space of all vectors orthogonal to either A or B. Note that a line not equal to zero cannot be orthogonal to A and B at the same time. What would be the space orthogonal to that space? It would be A ∩ B.
Conveniently, you can obtain an orthonormal basis of a set of linearly independent vectors as well as a basis for the space orthogonal to those vectors through a full QR factorization.