Proving the properties of the cross product 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?

In the first one you can argue that $(a*b)$ is perpendicular to both $a$ and $b$ so its dot product with either of them must be zero.
