Definition 1.3.1.
Let \(\vec{u}, \vec{v}, \vec{w}\) be vectors in either \(\R^2\) or \(\R^3\text{,}\) the dot product of \(\vec{u}\) and \(\vec{v}\) is defined as
\begin{equation*}
\begin{array}{cclr}
\vec{u} \cdot \vec{v} \amp=\amp u_1\,v_1 + u_2\, v_2 \hspace{1cm} \amp \text{ if } \vec{u}, \vec{v} \in \R^2\\
\amp \amp \amp\\
\amp = \amp u_1\,v_1 + u_2\, v_2 + u_3\,v_3 \hspace{1cm} \amp \text{ if } \vec{u}, \vec{v} \in \R^3
\end{array}
\end{equation*}