Here's a connection between (three dimensional) cross product and quaternion product, which involves nothing more than calculation. Its consequences are supposedly quite deep. And you'll need somebody who knows algebra to explain them, since I don't know or understand them. Among other things, it goes some way towards explaining WHY we have such a beautiful cross product of 2 vectors in R3, but nowhere else (except in R7, which uses octonions and is accordingly not as pretty).

So let's say we have 2 vectors in R3, say (u,v,w) and (x,y,z). Let's express them as quaternions in the stupidest possible manner:

ui + vj + wk, and xi + yj + zk
(Ever wonder WHY engineers like to call the 3 basis vectors i, j and k? There was a war in R3 between the quaternion people and the vector people; the vector people won, but not totally!).

Let's multiply them and see what we get:

-(ux+vy+wz) + (vz-wy)i + (uz-wx)j + (uy-vx)k
Voila! As if by magic, the coordinates on i,j and k are precisely those for cross product. And what's that ugly (real number) scalar doing there? It's just the scalar product of our vectors, with a change of sign.

No wonder so many surprising identities hold for cross product and scalar product -- we've just shown a deep algebraic connection...