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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?

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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.

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An account of multiplication of vectors, both scalar products and vector products.

Air Routes

Age 16 to 18
Challenge Level
The question asks for the use of 3D coordinates and scalar products in a very practical context.