Dot Product and the Law of Cosines

Place $\vec{u}=\overrightarrow{OA}$ and $\vec{v}=\overrightarrow{OB}$ at the origin $O$ so that $\overrightarrow{AB}=\vec{v}-\vec{u}=(x_2-x_1,y_2-y_1)$. By the Law of Cosines in $\triangle OAB$, the squared side length opposite angle $\theta$ is $|\vec{v}-\vec{u}|^2=|\vec{u}|^2+|\vec{v}|^2-2|\vec{u}|\,|\vec{v}|\cos\theta$.

On the other hand, computing the squared lengths in coordinates gives $|\vec{u}|^2=x_1^2+y_1^2$, $|\vec{v}|^2=x_2^2+y_2^2$, and $|\vec{v}-\vec{u}|^2=(x_2-x_1)^2+(y_2-y_1)^2=x_1^2+y_1^2+x_2^2+y_2^2-2(x_1x_2+y_1y_2)$.

Equating the two expressions for $|\vec{v}-\vec{u}|^2$ and subtracting $x_1^2+y_1^2+x_2^2+y_2^2$ from both sides yields $-2|\vec{u}|\,|\vec{v}|\cos\theta=-2(x_1x_2+y_1y_2)$. Dividing by $-2$ gives $\vec{u}\cdot\vec{v}=|\vec{u}|\,|\vec{v}|\cos\theta=x_1x_2+y_1y_2$.