Answer
The required value is $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=-\frac{9}{5}\mathbf{i}+\frac{18}{5}\mathbf{j}$.
Work Step by Step
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 $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=-\frac{9}{5}\mathbf{i}+\frac{18}{5}\mathbf{j}$.
