## Precalculus (6th Edition) Blitzer

$\mathbf{v}\cdot \mathbf{w}=0$ and $\mathbf{v}\cdot \mathbf{v}=25$
The dot product, $\mathbf{v}\cdot \mathbf{w}$ as, \begin{align} & \mathbf{v}\cdot \mathbf{w=}\left( 5\mathbf{i}+0\mathbf{j} \right)\cdot \left( 0\mathbf{i}+\mathbf{j} \right) \\ & =\left( 5 \right)\mathbf{i}\cdot \left( 0 \right)\mathbf{i+}\left( 5 \right)\mathbf{i}\cdot \left( 1 \right)\mathbf{j+}\left( 0 \right)\mathbf{j}\cdot \left( 0 \right)\mathbf{i+}\left( 0 \right)\mathbf{j}\cdot \left( 1 \right)\mathbf{j} \\ & =0\left( \mathbf{i}\cdot \mathbf{i} \right)+5\left( \mathbf{i}\cdot \mathbf{j} \right)+0\left( \mathbf{j}\cdot \mathbf{i} \right)+0\left( \mathbf{j}\cdot \mathbf{j} \right) \\ & =0\left( 1 \right)+5\left( 0 \right)+0+0 \end{align} Solve ahead to get the result as, \begin{align} & \mathbf{v}\cdot \mathbf{w}=0\left( 1 \right)+5\left( 0 \right)+0+0 \\ & =0 \end{align} The dot product, $\mathbf{v}\cdot \mathbf{v}$ as, \begin{align} & \mathbf{v}\cdot \mathbf{v=}\left( 5\mathbf{i}+0\mathbf{j} \right)\cdot \left( 5\mathbf{i}+0\mathbf{j} \right) \\ & =\left( 5 \right)\mathbf{i}\cdot \left( 5 \right)\mathbf{i+}\left( 5 \right)\mathbf{i}\cdot \left( 0 \right)\mathbf{j+}\left( 0 \right)\mathbf{j}\cdot \left( 5 \right)\mathbf{i+}\left( 0 \right)\mathbf{j}\cdot \left( 0 \right)\mathbf{j} \\ & =25\left( \mathbf{i}\cdot \mathbf{i} \right)+0\left( \mathbf{i}\cdot \mathbf{j} \right)+0\left( \mathbf{j}\cdot \mathbf{i} \right)+0\left( \mathbf{j}\cdot \mathbf{j} \right) \\ & =25\left( 1 \right)+0+0+0 \end{align} Solve ahead to get the result as, \begin{align} & \mathbf{v}\cdot \mathbf{v}=25\left( 1 \right)+0+0+0 \\ & =25 \end{align}