A challenge inspired by a test I took a month or so ago:

Given three vectors a, b, c in real Cartesion 3-space (three real number components), such that a ≠ 0, a × b = a × c, and a * b = a * c, where × is the exterior (cross) product, and * is the interior (dot) product, formulate three separate proofs of that b = c, without resorting to components.

All three proofs are quite simple, and two are fairly immediately obvious if you are familiar with V3 vectors, but the third is apparently less obvious, as my teacher said I'm the only person who he'd ever seen prove it that way.
If it's a hint, the third way isn't geometric (which is probably why it's not often seen, given that it's just as simple as the other two, if not more).