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Let $V, W$ be vectors in $\mathbb{R}^{n}$.

a) Show that the Pythagorean relation $\|V+W\|^{2}=\|V\|^{2}+\|W\|^{2}$ holds if and only if $V$ and $W$ are orthogonal.

b) Prove the parallelogram identity $\|V+W\|^{2}+\|V-W\|^{2}=2\|V\|^{2}+2\|W\|^{2}$ and interpret it geometrically. [This is true in any real inner product space].
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