## Precalculus (6th Edition) Blitzer

$\mathbf{v}\cdot \mathbf{w}=95$ and $\mathbf{v}\cdot \mathbf{v}=73$
The dot product, $\mathbf{v}\cdot \mathbf{w}$ as, \begin{align} & \mathbf{v}\cdot \mathbf{w=}\left( -8\mathbf{i}-3\mathbf{j} \right)\cdot \left( -10\mathbf{i}-5\mathbf{j} \right) \\ & =\left( -8 \right)\mathbf{i}\cdot \left( -10 \right)\mathbf{i+}\left( -8 \right)\mathbf{i}\cdot \left( -5 \right)\mathbf{j+}\left( -3 \right)\mathbf{j}\cdot \left( -10 \right)\mathbf{i+}\left( -3 \right)\mathbf{j}\cdot \left( -5 \right)\mathbf{j} \\ & =80\left( \mathbf{i}\cdot \mathbf{i} \right)+40\left( \mathbf{i}\cdot \mathbf{j} \right)+30\left( \mathbf{j}\cdot \mathbf{i} \right)+15\left( \mathbf{j}\cdot \mathbf{j} \right) \\ & =80\left( 1 \right)+40\left( 0 \right)+30\left( 0 \right)+15\left( 1 \right) \end{align} Solve ahead to get the result as, \begin{align} & \mathbf{v}\cdot \mathbf{w}=80\left( 1 \right)+40\left( 0 \right)+30\left( 0 \right)+15\left( 1 \right) \\ & =80+0+0+15 \\ & =95 \end{align} The dot product, $\mathbf{v}\cdot \mathbf{v}$ as, \begin{align} & \mathbf{v}\cdot \mathbf{v=}\left( -8\mathbf{i}-3\mathbf{j} \right)\cdot \left( -8\mathbf{i}-3\mathbf{j} \right) \\ & =\left( -8 \right)\mathbf{i}\cdot \left( -8 \right)\mathbf{i+}\left( -8 \right)\mathbf{i}\cdot \left( -3 \right)\mathbf{j+}\left( -3 \right)\mathbf{j}\cdot \left( -8 \right)\mathbf{i+}\left( -3 \right)\mathbf{j}\cdot \left( -3 \right)\mathbf{j} \\ & =64\left( \mathbf{i}\cdot \mathbf{i} \right)+24\left( \mathbf{i}\cdot \mathbf{j} \right)+24\left( \mathbf{j}\cdot \mathbf{i} \right)+9\left( \mathbf{j}\cdot \mathbf{j} \right) \\ & =64\left( 1 \right)+24\left( 0 \right)+24\left( 0 \right)+9\left( 1 \right) \end{align} Solve ahead to get the result as, \begin{align} & \mathbf{v}\cdot \mathbf{v}=64\left( 1 \right)+24\left( 0 \right)+24\left( 0 \right)+9\left( 1 \right) \\ & =64+0+0+9 \\ & =73 \end{align}