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How did they go from 2b⋅c to -2bccosA? Where did they get the negative sign from?
2 Answers
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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.
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Notice that the vector $\vec{b}$ points into the vertex $A$ whereas $\vec{c}$ points out. Thus, we apply the formula for the dot-product in terms of the interior angle between $-\vec{b}$ and $\vec{c}$ hence $-\vec{b} \cdot \vec{c} = -bc\cos A$
