Section 1.9 The Law of Cosines

The definition of the dot product can be used to prove several familiar formulas. For example, consider Figure 1.9.1, in which \(\CC=\BB-\AA\text{.}\) Then
\begin{align*}
\CC\cdot\CC \amp = (\BB -\AA) \cdot (\BB -\AA) \nonumber\\
\amp = \AA\cdot\AA + \BB\cdot\BB - 2 \,\AA\cdot\BB
\end{align*}
or equivalently
\begin{equation}
\,|\CC|^2 = |\AA|^2 + |\BB|^2 - 2|\AA||\BB|\cos\theta\tag{1.9.1}
\end{equation}
which is just the Law of Cosines.