So I learned that if a vector (a1, a2, a3) is transformed to a different set of coordinates, and the components of the resulting vector are squared like so: (a1'2, a2'2, a3'2), this result is not itself a vector. The proof for this simply shows that each component ai'2 does not transform to ai2 when brought to the original coordinates. I'm having a lot of trouble understanding this. Say the vector (1,0,0) is transformed by a 90O rotation about the z-axis. It becomes (0,-1,0). The "square" of this is (0,1,0). Sure this doesn't transform back to (1,0,0) but if this triplet is considered in isolation, without any reference to the original vector, there doesn't seem to be any reason not to consider it a vector.