I have to prove the following properties
- $(a \times b) \cdot a = (a \times b) \cdot b = 0$
- $|a \times b|^2 = |a|^2 |b|^2 - (a \cdot b)^2$
- $(a \times b) \times c = (a \cdot c)b - (b \cdot c)a$
The way I am proving them is by literally saying:
$$a = (A, B, C)$$ $$b = (D, E, F)$$ $$c = (G, H, I)$$
and then just showing both sides of the equals sign are the same. Is this the best way or is there a quicker way?