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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?

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.

### Air Routes

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.

# Quaternions and Reflections

##### Age 16 to 18 Challenge Level:
The 'recipe' is given in the question. You need to know that the scalar product of two perpendicular vectors is 0 so that, if one vector lies in a plane and the other is normal to the plane, then their scalar product is zero. This gives the equation of a plane through the origin in ${\bf R^3}$ as $v\cdot n = a x + b y + c z = 0$. The diagram should help you to visualise that, if $u_0$ is on the plane and $n$ is a vector normal to the plane, then the points $u_0 + t n$ and $u_0 - t n$ are reflections of each other in the plane.

Where quaternions are equivalent to vectors we are not using boldface fonts other than in introducing the unit vectors ${\bf i, j, k}$ along the axes in ${\bf R^3}$.

The quaternion functions and quaternion algebra give a neat and efficient way to work with reflections in ${\bf R^3}$ and they are very useful in computer graphics programs.