Precalculus (6th Edition) Blitzer

The required value is $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=-\frac{9}{5}\mathbf{i}+\frac{18}{5}\mathbf{j}$.
We have \begin{align} & \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=\frac{\mathbf{v}\cdot \mathbf{w}}{{{\left\| \mathbf{w} \right\|}^{2}}}\mathbf{w} \\ & =\frac{\left( -5\mathbf{i}+2\mathbf{j} \right)\cdot \left( 2\mathbf{i}-4\mathbf{j} \right)}{{{\left( \sqrt{{{2}^{2}}+{{\left( -4 \right)}^{2}}} \right)}^{2}}}\left( 2\mathbf{i}-4\mathbf{j} \right) \\ & =\frac{\left( -5\left( 2 \right) \right)+2\left( -4 \right)}{{{\left( \sqrt{20} \right)}^{2}}}\left( 2\mathbf{i}-4\mathbf{j} \right) \\ & =-\frac{18}{20}\left( 2\mathbf{i}-4\mathbf{j} \right) \end{align} So, \begin{align} & \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=-\frac{9}{10}\left( 2\mathbf{i}-4\mathbf{j} \right) \\ & =-\frac{9}{5}\mathbf{i}+\frac{18}{5}\mathbf{j} \end{align} Hence, $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=-\frac{9}{5}\mathbf{i}+\frac{18}{5}\mathbf{j}$ Therefore, the required value is $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=-\frac{9}{5}\mathbf{i}+\frac{18}{5}\mathbf{j}$.