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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.

# Cross with the Scalar Product

##### Stage: 5 Challenge Level:

Always remember the key geometrical facts about scalar and cross products for non-zero vectors ${\bf u}$ and ${\bf v}$:

1) ${\bf u}\cdot {\bf v}=0$ if and only if ${\bf u}$ and ${\bf v}$ are perpendicular.

2) ${\bf u}\times {\bf v}=0$ if and only if ${\bf u}$ and ${\bf v}$ are parallel.

3) ${\bf u}\times {\bf v}$ is perpendicular to both ${\bf u}$ and ${\bf v}$.

This problem uses mathematical concepts covered in the later Further Pure Mathematics A Level Modules.