## Section3.6Dot 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 $$\vec{v}\perp \vec{w}\Leftrightarrow \vec{v}\cdot\vec{w}=0\text{.}$$

### Activity3.6.1.

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).

### Activity3.6.2.

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.