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Flexi Quads

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?

Flexi Quad Tan

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

Age 16 to 18 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.