Prove a given vector identity

by RoRi

April 27, 2016

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 $\lVert A + B \rVert = \lVert A - B \rVert. \qquad \blacksquare$

Stumbling Robot

Prove a given vector identity

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