In this question we see how quaternions are used to give rotations of ${\bf R^3}$. 
(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 $2\theta$.
