# Chapter 6 - Section 6.7 - The Dot Product - Exercise Set - Page 792: 1

$\mathbf{v}\cdot \mathbf{w}=6$ and $\mathbf{v}\cdot \mathbf{v}=10$

#### Work Step by Step

The dot product, $\mathbf{v}\cdot \mathbf{w}$ as, \begin{align} & \mathbf{v}\cdot \mathbf{w=}\left( 3\mathbf{i+j} \right)\cdot \left( \mathbf{i+}3\mathbf{j} \right) \\ & =3\mathbf{i}\cdot \mathbf{i+}3\mathbf{i}\cdot 3\mathbf{j+j}\cdot \mathbf{i+j}\cdot 3\mathbf{j} \\ & =3\left( 1 \right)+9\left( \mathbf{i}\cdot \mathbf{j} \right)+\left( \mathbf{j}\cdot \mathbf{i} \right)+3\left( \mathbf{j}\cdot \mathbf{j} \right) \\ & =3+9\left( 0 \right)+\left( 0 \right)+3\left( 1 \right) \end{align} Solve ahead to get the result as, \begin{align} & \mathbf{v}\cdot \mathbf{w}=3+9\left( 0 \right)+\left( 0 \right)+3\left( 1 \right) \\ & =3+0+0+3 \\ & =6 \end{align} The dot product, $\mathbf{v}\cdot \mathbf{v}$ as, \begin{align} & \mathbf{v}\cdot \mathbf{v=}\left( 3\mathbf{i+j} \right)\cdot \left( 3\mathbf{i+j} \right) \\ & =3\mathbf{i}\cdot 3\mathbf{i+}3\mathbf{i}\cdot \mathbf{j+j}\cdot 3\mathbf{i+j}\cdot \mathbf{j} \\ & =9\left( \mathbf{i}\cdot \mathbf{i} \right)+3\left( \mathbf{i}\cdot \mathbf{j} \right)+3\left( \mathbf{j}\cdot \mathbf{i} \right)+\left( \mathbf{j}\cdot \mathbf{j} \right) \\ & =9\left( 1 \right)+3\left( 0 \right)+3\left( 0 \right)+1 \end{align} Solve ahead to get the result as, \begin{align} & \mathbf{v}\cdot \mathbf{v}=9\left( 1 \right)+3\left( 0 \right)+3\left( 0 \right)+1 \\ & =9+0+0+1 \\ & =10 \end{align}

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