i, j, k are unit quaternions. . is the dot product. (Perhaps you might want to look at this as a "formal" or "algebraic" dot product, in that in my construction above, you're taking a product between R^3 and H^3, mapping to H.) If v = <x, y, z>, then v . <i, j, k> = xi + yj + zk. So yes, I simply mean a vector. But we are throwing around different concepts (vectors in R^3, quaternions, mapping between the two, etc.) with different notions of "multiplication" (cross, dot, rotation, quaternion multiplication), and I sought to make all those clear. Robert On Thu, Jun 26, 2014 at 4:04 PM, Henry Baker <hbaker1@pipeline.com> wrote:

Sorry, Robert, but I still don't understand.

What is the meaning of "(v . <i, j, k>)" ?

Do you simply mean a vector (= pure (imaginary) quaternion; i.e., S(v)=0) ?

At 01:50 PM 6/26/2014, Robert Smith wrote:

...

Most formally speaking, for a quaternion Q and a vector v in R^3, and for * meaning quaternion multiplication,

Q # v = Q * (v . <i, j, k>) * Q^{-1}

where i, j, and k are the pure unit quaternions. Originally I said conj(Q) instead of Q^{-1} which is only correct some of the time.

Robert

On Thu, Jun 26, 2014 at 11:23 AM, Henry Baker <hbaker1@pipeline.com> wrote:

...

I didn't understand your "#" operation. Could you please try again?