Section 3.6 Dot Products
The geometric, coordinate independent definition of the dot product of two vectors \(\vec{v}\) and \(\vec{w}\text{,}\) thought of as arrows in space, is
\begin{equation}
\vec{v}\cdot\vec{w}=\vert \vec{v}\vert \vert \vec{w}\vert \cos{\gamma}\tag{3.6.1}
\end{equation}
where \(\gamma\) is the angle between \(\vec{v}\) and \(\vec{w}\text{.}\) If we represent the two vectors by their components in a coordinate system
\begin{equation}
\vec{v}\doteq\begin{pmatrix}
v_x\\
v_y\\
v_z
\end{pmatrix}
\qquad\qquad
\vec{w}\doteq\begin{pmatrix}
w_x\\
w_y\\
w_z
\end{pmatrix}\tag{3.6.2}
\end{equation}
then the geometric definition is equivalent to the algebraic definition
\begin{align}
\vec{v}\cdot\vec{w}
\amp =\begin{pmatrix}v_x\amp v_y\amp v_z\end{pmatrix}
\begin{pmatrix}w_x\\w_y\\w_z\end{pmatrix}\tag{3.6.3}\\
\amp = v_x w_x+ v_y w_y + v_z w_z\tag{3.6.4}
\end{align}
In physics, the dot product is most often used for two purposes: to find the length of a vector \(\vert \vec{v}\vert = \sqrt{\vec{v}\cdot\vec{v}}\) and to show that two vectors are perpendicular to each other \(\vec{v}\perp \vec{w}\Leftrightarrow \vec{v}\cdot\vec{w}=0\text{.}\)
Activity 3.2.
Show, from the geometric definition of the dot product (3.6.1), that the two vectors are perpendicular if and only if their dot product is zero \(\vec{v}\perp \vec{w}\Leftrightarrow \vec{v}\cdot\vec{w}=0\text{.}\) Repeat for the algebraic definition (3.6.4).The generalization of the dot product to vectors with complex components can be found in Section 3.7.
