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(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.

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?
As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.
Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.