Q:

Deriving the cosine formula using vectors?

How did they go from 2b⋅c to -2bccosA? Where did they get the negative sign from?

A:

The problem is that $b$ and $c$ do not point in the 'same' direction. The scalar product of $b$ and $c$ is proportional to the angle between $b$ and $c$, but here the angle $A$ is not between $b$ and $c$ but rather the supplementary angle.

