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Prove an identity for two vectors in Euclidean n-space

by RoRi

For two vectors A, B \in \mathbb{R}^n prove that we have the identity

    \[ \lVert A + B\rVert^2 + \lVert A - B \rVert^2 = 2 \lVert A \rVert^2 + 2 \lVert B \rVert^2. \]

What geometric theorem can you deduce from this identity?

Proof. We compute

    \begin{align*} \lVert A + B \rVert^2 + \lVert A - B \rVert^2 &= \sum_{i=1}^n (a_i + b_i)^2 + \sum_{i=1}^n (a_i - b_i)^2 \\[9pt] &= \sum_{i=1}^n (2a_i^2 + 2b_i^2) \\[9pt] &= 2 \sum_{i=1}^n a_i^2 + 2 \sum_{i=1}^n b_i^2 \\[9pt] &= 2 \lVert A \rVert^2 + 2 \lVert B \rVert^2. \qquad \blacksquare \end{align*}

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