Why quaternions flopped
After last week's post about the relationship between quaternions and the dot and cross products of vectors, I was asked why I thought quaternions didn't end up being very useful. I'd guess that the short answer is that they really don't model the real world very well.
It definitely isn't because multiplication of quaternions isn't commutative. Matrix multiplication also isn't commutative, yet matrices have ended up being very useful.
Instead, I'd guess that the problem with quaternions is that calculus doesn't work very well with them. With the complex numbers, there's a nice, clean way to characterize analytic functions: df/dz* = 0. But that doesn't seem to extend to the quaternions very well.
If we have a quaternion
q = q0 + q1 i + q2 j + q3 k
and its conjugate
q* = q0 - q1 i – q2 j - q3 k
then we have
qq* = q02 + q12 + q22 + q32
which looks about right.
But we also find that we can write
q* = -(q + i q i + j q j + k q k) / 2
And because q and q* are connected in this way, we can never find a nice condition like df/dq* = 0 to characterize functions of a quaternion variable like we can for analytic functions. So I'd guess that's also part of why they haven't ended up being very useful.