Vectors - theory on cross product

Question

If $X$ is a point on a line through $P$ and $Q$, $X=OX, P=OP, Q=OQ$ (all are vectors but $X$) then:

$$X \times (Q-P) = P \times Q$$

I subbed in the $OX$, etc and simplified, but did not get each side equal to each other, how would I solve this?

Answer 1

Consider what it means for $X$ to lie on the line through $P$ and $Q$.

This means that there is some $t$ such that $X = P+(Q-P)t$ (since $P+(Q-P)t$ parameterizes this line).

Now you have

$$X \times (Q-P) = [P+(Q-P)t] \times (Q-P)$$

Next, use the fact that the cross product is "bilinear" (i.e. obeys distributive type laws and you can pull out scalars) as well as the fact that anything cross itself is zero.