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?
 A: It depends on what you mean is the best way. At times with vectors, there isn't a need to resort to horrendous amounts of computations, if you understand what is happening. For example, to show that 2 vectors $a, b$  are the same, one way of doing that is to show that $a \cdot v = b \cdot v$ for all vectors $v$, in particular the unit vectors $i, j, k$.
Here are some suggestions


*

*Prove by the geometric interpretation of cross and dot product. The cross product produces a vector that is orthogonal to both $a$ and $b$, hence the dot product is 0.

*Prove by the trigonometric identity, use Pythagorean Theorem.

*Prove by considering the $x, y, z$ coordinates separately.
A: 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.
A: In the 2nd one u can use the fact that $|a*b|^2 = |a|^2|b|^2(sin \theta)^2$ and similarly for dot product i.e. $|a.b|^2 = |a|^2|b|^2(cos \theta)^2$,$\theta$ being the angle between a,b.
