State and prove the parallelogram law?
Solution 1:
\begin{align*} |\mathbf{u}+\mathbf{v}|^{2}+|\mathbf{u}-\mathbf{v}|^{2} &= (\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}+\mathbf{v})+ (\mathbf{u}-\mathbf{v}) \cdot (\mathbf{u}-\mathbf{v}) \\ &= \mathbf{u} \cdot \mathbf{u}+2\mathbf{u} \cdot \mathbf{v}+ \mathbf{v} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{u}- 2\mathbf{u} \cdot \mathbf{v}+\mathbf{v} \cdot \mathbf{v} \\ &= 2(\mathbf{u} \cdot \mathbf{u}+\mathbf{v} \cdot \mathbf{v}) \\ &= 2(|\mathbf{u}|^{2}+|\mathbf{v}|^{2}) \end{align*}