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What is the geometric interpretation of $\|a-b\|^2+\|a+b\|^2=2\|a\|^2 + 2\|b\|^2$? I thought something with a triangle, but I couldn't picture that out with drawing the vectors.

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Use $\backslash|$ rather than $||$. – Did Dec 21 '11 at 11:40
up vote 3 down vote accepted

I have the answer below, but just noticed the homework tag. Thinking of a triangle is close..

If you just want hints: What four sided figure would $a$ and $b$ determine? And, how do the vectors $a+b$ and $a-b$ relate to this figure?

The sum of the squares of the lengths of the diagonals of a parallelogram is equal the sum of the squares of the lengths of the sides of the parollelogram.

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To prove this analytically, write the squares of the norms in terms of the inner product: $||x||^2=x\cdot x$. – David Mitra Dec 21 '11 at 11:31
i think it is called the apollonius' theorem? – abel Dec 14 '14 at 0:00

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