Dot Products

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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 3.1.

Show, from the geometric definition of the dot product (3.6.1), that the length of a vector is given by \(\vert \vec{v}\vert = \sqrt{\vec{v}\cdot\vec{v}}\text{.}\) Repeat for the algebraic definition (3.6.4).

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.