Moreover, quaternions do not rotate a vector by multiplication: they do it by conjugation (i.e. q*v*q^{-1} ).

Allow me to step in here for a short moment. This has always been unintuitive to me, and though not understanding (other than "because the formulas work out") why it

*must *work that way didn't prevent me from using quaternions (you can likely grow old and never have a

*real need *to know why it works!), it is still interesting.

I've had a mathematician explain (or *try to *explain) that reason to me not long ago. And while I don't claim that I find quaternions or 4D space much more intuitive now (though funnily, homogeneous coordinates are 4D too, and they're entirely intuitive), it is much clearer to me now why you rotate **qvq**^{-1} with **θ/2** rather than just **qv** with **θ**.

Unit quaternions do not rotate in 3D at all, but they perform two "isoclinic" rotations in 4D (let's pretend you can *truly *imagine how that looks like, I can't for sure). But the important detail is that 3D space is a plane in 4D (or so the mathematicians tell us, let's trust them!), and one of these rotations is the rotation in the 3D plane that we want, the other is... something else, a rotation in a plane that is orthogonal to the 3D plane. Therefore, if you just rotated using *one *quaternion which corresponds to **θ**, then you would get... *some transformation*, but not a proper rotation in 3D, which you want.

If you multiply on the left, both rotations go in the same sense, if you multiply on the right, they go in the respective opposite sense (so one is clockwise, and the other counter-clockwise), and a quaternion's conjugate corresponds to the same orientation, but rotated the other way around.

So the trick is to build a quaternion that only rotates **θ/2** and do one multiplication on the left, and one on the right with the conjugate, which does the second half of the 3D rotation that you want and at the same time "unrotates" the other one that you don't want.