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Prove a given vector identity

Prove that for vectors A, B \in \mathbb{R}^n we have the identity

    \[ \lVert A + B \rVert^2 - \lVert A - B \rVert^2 = 4A \cdot B. \]

Hence, A \cdot B = 0 if and only if

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


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 4a_i b_i \\  &= 4 \sum_{i=1}^n a_i b_i \\  &= 4 A \cdot B. \end{align*}

Thus, A \cdot B = 0 if and only if \lVert A + B \rVert = \lVert A - B \rVert. \qquad \blacksquare

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