Show that Two Vectors Making Supplementary Angles?

Question

I just need a start. I am not looking for whole prove, but it'd be more appreciated if I get one.

Q. Use Theorem u . v = |u| |v| cos a and the trigonometric identity, cos (180-a) = -cos a, to conclude that the angles formed by u and v and by -u and v are supplementary (add to pi rad).

a link to the Theorem

Answer 1

We have $u \cdot v = |u||v|cos(a)$ and $(-u) \cdot v = -(u \cdot v) = |u||v|cos(b) $

Rearranging gives us $cos(a) = \frac{u \cdot v}{|u||v|}$ and $cos(b) = -\frac{u \cdot v}{|u||v|}$

Then $cos(a)=-cos(b) \Rightarrow cos(b)=-cos(a) \Rightarrow cos(b)=cos(\pi -a)$