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

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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].
in Mathematics by ♦ Platinum (164,236 points) | 256 views

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