Quaternions and Reflections
Age 16 to 18 Challenge Level:
Quaternions are 4dimensional numbers of the form $(a,x,y,z)=
a+x{\bf i}+y{\bf j}+z{\bf k}$ where $a, x, y$ and $z$ are real
numbers, ${\bf i, j}$ and ${\bf k}$ are all different square roots
of $1$ and ${\bf i j} = {\bf k} = {\bf j i},\ {\bf j k} = {\bf i}
= {\bf k j},\ {\bf k i} = {\bf j} = {\bf i k}.$
The quaternion $a + x{\bf i} + y{\bf j} + z{\bf k}$ has a real part
$a$ and a
pure quaternion
part $x{\bf i} + y{\bf j}+ z{\bf k}$ where ${\bf i, j}$, and ${\bf
k}$ are unit vectors along the axes in ${\bf R^3}$.
(1) For the pure quaternions $v_1 = x_1{\bf i}+y_1{\bf j} + z_1{\bf
k}$ and $v_2 = x_2{\bf i} +y_2{\bf j} +z_2{\bf k}$ evaluate the
quaternion product $v_1v_2$ and compare your answer to the scalar
and vector products $v_1 \cdot v_2$ and $v_1 \times v_2$.
(2) Evaluate the quaternion product $v^2$ where $v=x{\bf i} + y{\bf
j} + z{\bf k}$ and $v = \sqrt (x^2 + y^2 + z^2) = 1$.
Show that, for all real angles $\theta$ and $\phi$, $$v = \cos
\theta \cos \phi {\bf i} + \cos \theta \sin \phi {\bf j} + \sin
\theta {\bf k}$$ is a square root of 1. This gives the set of all
the points on the unit sphere in ${\bf R^3}$ and shows that the
quaternion $1$ has infinitely many square roots (which we call
unit pure quaternions ).

(3) Take any unit pure quaternion $n$ ($n^2=1$) and consider
the plane $\Pi$ through the origin in ${\bf R^3}$ with normal
vector $n$. Then the plane $\Pi$ has equation $a x + b y + c z = 0
= v\cdot n$.
If $u_0$ is a point on the plane $\Pi$ then $u_0\cdot n =0$
and the points $u_0+ t n$ and $u_0  t n$ are reflections of each
other in the plane.

Show that the quaternion map $F(u) = n u n$ gives reflection
in the plane $\Pi$ by showing:
(i)$u_0n = n u_0$ and hence $F(u_0)=u_0$ so that all points
on the plane are fixed by this mapping, and
(ii) $F(u_0 + t n) = u_0  t n$ for all scalars $t$.
If you want to know how
quaternions are used in computer graphics and animation in film
making read the Plus Article Maths
goes to the movies .