(Solved): Consider a parallelogram. Choose a vertex and call it A. Let u and v be the two adjacent sides of t ...

Consider a parallelogram. Choose a vertex and call it $A$. Let $\mathbf{u}$ and $\mathbf{v}$ be the two adjacent sides of the parallelogram that have their tail at $A$. Then do the following: (a) Prove the following fact using vector algebra: $\Vert \mathbf{u}+\mathbf{v}{\Vert }^2+\Vert \mathbf{u}-\mathbf{v}{\Vert }^2=2\Vert \mathbf{u}{\Vert }^2+2\Vert \mathbf{v}{\Vert }^2$. Interpret this result geometrically by stating what it says about parallelograms. (b) Prove that $\mathbf{u}\cdot \mathbf{v}=\frac{1}{4}\Vert \mathbf{u}+\mathbf{v}{\Vert }^2-\frac{1}{4}\Vert \mathbf{u}-\mathbf{v}{\Vert }^2$.