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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?

### Three by One

There are many different methods to solve this geometrical problem - how many can you find?

### Tetra Perp

Show that the edges AD and BC of a tetrahedron ABCD are mutually perpendicular when: AB²+CD² = AC²+BD².

# Quaternions and Rotations

##### Age 16 to 18 Challenge Level:
Comparing the results in parts (1) and (2) we see that in part (1) $q = \cos 45^o + \sin 45^o{\bf i}$ and the map $qvq^{-1}$ fixes the x-axis and gives a rotation of twice 45 degrees about the x-axis.

The result in (2) is slightly more general showing that where $q = \cos \theta + \sin \theta {\bf k}$ the map $q v q^{-1}$ fixes the z-axis and gives a rotation of $2\theta$ about the z-axis. \par For a more general account of quaternions and rotations see the article....