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Prove that if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n}$ with the Euclidean inner product, then $$\mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2}$$
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Proof

$\|\mathbf{u}+\mathbf{v}\|^{2}=(\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=\|\mathbf{u}\|^{2}+2(\mathbf{u} \cdot \mathbf{v})+\|\mathbf{v}\|^{2}$
$\|\mathbf{u}-\mathbf{v}\|^{2}=(\mathbf{u}-\mathbf{v}) \cdot(\mathbf{u}-\mathbf{v})=\|\mathbf{u}\|^{2}-2(\mathbf{u} \cdot \mathbf{v})+\|\mathbf{v}\|^{2}$

by Platinum (122,080 points)

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