I don't think so. You have to use more information about the cross product than just the magnitudes. Counterexample to this claim:If that conclusion can be made, is it sufficient to show that if two vectors have the same magnitude and that the angles between them and the vector are the same, then they are the same vector?

The dot products are all zero because of orthogonality. The magnitudes are all the same, and the angle between a and c is the same as between b and c: 90 degrees. But a is not equal to b.

You have to use all the hypotheses: the dot products are equal, the vector products are all equal (and not just the magnitudes, but the directions of the vector products are equal!), and you have to use the fact that c is nonzero.

I would probably go long here, and starting writing out equations component-wise, and see where I get.