Question

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

Answer 1

Proof

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