Using a left-handed coordinate system, let
Q = axisAngle({0,0,1}, 1/4*pi) * axisAngle({0,1,0}, 1/4*pi)
be the quaternion representing the rotation "1/8 circle clockwise around the global Z axis, then inclinate 1/8 circle down". Also, let $V$ be the result of rotating $\{1,0,0\}$ by $Q$. By using a calculator, I found that:
$V = \left\{\frac12, \frac12, -\sqrt{\frac12}\right\}$
But I expected $V$ to be:
$V = \left\{\sqrt{\frac13}, \sqrt{\frac13}, -\sqrt{\frac13}\right\}$
After all, if we apply said rotation manually, that's the direction we get. What is wrong with my elaboration?
V = Q * (0,Vx,Vy,Vz) * conjugate(Q)
, whereconjugate (Qw,Qx,Qy,Qz) = (Qw,-Qx,-Qy,-Qz)
. I'm not sure what you mean, both V's are of unit length. $\endgroup$