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Geometry Level 3

In a parallelopiped the ratio of the sum of the squares on the four diagonals to the sum of the squares on the three coterminous edges is?

This question is a part of Killer.......vectors .

1 5 infinite 2 4 3

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Michael Mendrin
Apr 25, 2015

A cube is as good as a parallelopiped as any

Nevertheless, this is interesting. It's an example of an invariant under affine transforms. Invariants play an important role in physics.

Probably the quickest way to work this out in the general case is to consider vectors $v1, v2, v3$ . Then

$4\left( v1\cdot v1+v2\cdot v2+v3\cdot v3 \right) =$

$\left( v1+v2+v3 \right) \cdot \left( v1+v2+v3 \right) +$ $\left( -v1+v2+v3 \right) \cdot \left( -v1+v2+v3 \right) +$ $\left( v1-v2+v3 \right) \cdot \left( v1-v2+v3 \right) +$ $\left( v1+v2-v3 \right) \cdot \left( v1+v2-v3 \right)$

Hence the ratio $4$

Kill Him!!!!!!!!!

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