Quaternions and Rotations

(1) Consider the quaternion $$q = {1\over \sqrt 2} + {1\over \sqrt 2}{\bf i} + 0{\bf j} + 0 {\bf k}.$$ (a) Show that the multiplicative inverse of $q$ is given by $$q^{-1} = {1\over \sqrt 2} - {1\over \sqrt 2}{\bf i}$$ (b) Show that for all scalar multiples $x = t{\bf i}$ of the vector ${\bf i}$, $q x = x q$ and hence $q x q^{-1} = x$. This proves that the map $F(x) = q x q^{-1}$ fixes every point on the x axis.

(c) What happens to points on the y axis under the mapping $F$? To answer this work out $F({\bf j})$. Also compute $F({\bf k})$ and show that ${\bf k} \to {\bf -j}.$

(2) Consider the quaternion $q = \cos \theta + \sin \theta {\bf k}$

(a) Show that $\cos \theta - \sin \theta {\bf k}$ is the multiplicative inverse of $q$.

(b) Show that $q{\bf k}q^{-1}={\bf k}$.

(c) Show that $$q v q^{-1}= r(\cos (2\theta + \phi) {\bf i} + \sin (2\theta + \phi){\bf j})$$ where $v = (r\cos \phi {\bf i} + \sin \phi {\bf j}+0{\bf k})$ and hence that the map $G(v)= q v q^{-1}$ is a rotation about the z axis by an angle $2\theta$.