## Precalculus (6th Edition) Blitzer

If $v$ and $w$ are non-zero vectors and $\theta$ is the smallest non-negative angle between $\mathbf{u}$ and $\mathbf{w}$, Then the angle between two vectors can be calculated as: \begin{align} & \cos \theta =\frac{\mathbf{u}\centerdot \mathbf{w}}{\left\| \mathbf{u} \right\|\left\| \mathbf{w} \right\|} \\ & \mathbf{u}\centerdot \mathbf{w}=\left\| \mathbf{u} \right\|\left\| \mathbf{w} \right\|\cos \theta \\ \end{align} Two vectors are said to be orthogonal if the angle between them is ${{90}^{\circ }}$ If $\mathbf{u}$ and $\mathbf{w}$ are orthogonal then, \begin{align} & \mathbf{u}\centerdot \mathbf{w}=\left\| \mathbf{u} \right\|\left\| \mathbf{w} \right\|\cos 90{}^\circ \\ & =\left\| \mathbf{u} \right\|\left\| \mathbf{w} \right\|\left( 0 \right) \\ & =0 \end{align} $\mathbf{u}\centerdot \mathbf{w}=0$