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 **R**^{3}, but nowhere else (except in **R**^{7}, which uses octonions and is accordingly not as pretty).

So let's say we have 2 vectors in **R**^{3}, 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

**R**^{3} 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...