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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 $2\theta$.

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?