Watch the video below for a proof of the law of cosines using vectors.
Section 3.1 Law of Cosines and the Angle between Two Vectors
Consider two vectors \(\mathbf u\) and \(\mathbf v\text{,}\) and let \(\mathbf w = \mathbf u - \mathbf v\text{,}\) as shown in Figure 3.1. By the law of cosines,
\begin{equation}
|\mathbf w|^2 = |\mathbf u|^2 + |\mathbf v|^2 - 2|\mathbf u||\mathbf v|\cos\theta.\tag{3.1}
\end{equation}
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Proof.
Writing \(\mathbf u = \langle u_1, u_2, u_3\rangle\) and \(\mathbf v = \langle v_1, v_2, v_3\rangle\text{,}\) note that \(\mathbf w = \mathbf u - \mathbf v\text{,}\) and hence
\begin{equation}
\begin{aligned}
|\mathbf w|^2 \amp = |\mathbf u - \mathbf v|^2 = (u_1-v_1)^2 + (u_2-v_2)^2 + (u_3-v_3)^2 \\
\amp = (u_1^2+u_2^2+u_3^2) + (v_1^2+v_2^2+v_3^2) - 2(u_1v_1+u_2v_2+u_3v_3).
\end{aligned}\tag{3.2}
\end{equation}
Also, from the law of cosines in (3.1) we have
\begin{equation}
|\mathbf w|^2 = (u_1^2+u_2^2+u_3^2) + (v_1^2+v_2^2+v_3^2) - 2|\mathbf u||\mathbf v|\cos\theta.\tag{3.3}
\end{equation}
From (3.2) and (3.3), it follows that \(u_1v_1+u_2v_2+u_3v_3 = |\mathbf u||\mathbf v|\cos\theta\text{,}\) and therefore the angle between the two vectors \(\mathbf u\) and \(\mathbf v\) can be evaluated using the following equation.
Theorem 3.3. Angle between Two Vectors.
\begin{equation}
\cos\theta = \frac{u_1v_1+u_2v_2+u_3v_3}{|\mathbf u||\mathbf v|}
\quad\iff\quad
\theta = \cos^{-1}\left(\frac{u_1v_1+u_2v_2+u_3v_3}{|\mathbf u||\mathbf v|}\right).\tag{3.4}
\end{equation}
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