1. ## Parallelogram

(1) Prove that the diagonals of an equilateral parallelogram are perpendicular.

So this is basically a square right? Is this essentially the same thing as $(A-B) \cdot (A+B) = 0$ where both of these vectors give the diagonals?

(2) Prove the law of sines using the cross product.

So we want to prove that $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

So $A \times B = |A||B| \sin \theta$ gives the area of a parallelogram formed by $A,B,C$. How would I proceed from here?

2. (1) Realize that if A & B are vectors that are adjacent sides of a parallelogram then (A+B) and (A-B) are the diagonals. We know $\left( {A + B} \right) \cdot \left( {A - B} \right) = \left\| A \right\|^2 - \left\| B \right\|^2$. Recalling what equilateral means the result follows at once.