Quaternions and spatial rotation

Remarks

For each unit quaternion $q$, the map $R_q$ is an element of the special orthogonal group $SO(3)$. If $q = \cos(\theta/2) + \sin(\theta/2)(u_1 i + u_2 j + u_3 k)$ with $(u_1, u_2, u_3)$ a unit vector, then $R_q$ is the rotation by $\theta$ about the axis $\R u$. One has $R_{pq} = R_p \circ R_q$, and the map $q \mapsto R_q$ from the unit quaternions onto $SO(3)$ is a surjective group homomorphism with kernel $\set{1, -1}$.

Example: $q = (1 + k)/\sqrt{2}$ has $\abs{q} = 1$ and $\bar q = (1 - k)/\sqrt{2}$. Then $R_q(i) = \tfrac12 (1 + k)\, i\, (1 - k) = \tfrac12 (i + j)(1 - k) = \tfrac12 (i + j + j - i) = j$, and $R_q(k) = k$. So $R_q$ is the quarter turn about the $z$-axis, as the formula with $\theta = \pi/2$ predicts.

Non-example: $q = 2$ is not a unit quaternion, and $v \mapsto q v \bar q = 4 v$ is a scaling, not a rotation. For any nonzero $q$ the map $v \mapsto q v q^{-1}$ equals $R_{q / \abs{q}}$, so one may also use non-unit quaternions in that form.