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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.