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Coordinated Crystals
Age 16 to 18
Challenge Level 
Focus on the atom at the origin and its nearest neighbours to begin
with.
Use the scalar product to work out the angles once you have visualised the structure.
Use the scalar product to work out the angles once you have visualised the structure.
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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?
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.