When it comes to planes, you usually would be describing them as vectors anyway... specifically, each plane is described by a vector which projects orthonormally from its own surface. If the vectors projecting from two planes are perpendicular, then by definition so are the planes. There is a direct equivalence. In any case you would use the fact that two vectors are perpendicular if, and only if their dot-product is zero. As you said in your post, there are infinite solutions because of rotations. The only way to eliminate that is to impose another constraint. The rule is: for the number N dimensions you are working in, you need N+1 constraining parameters to be able to solve for the vector/plane you want. I can tell you a general approach, if you like, but it requires drawing for clarity. e-mail me if you are interested.
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