Answer
The projection vector, $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=\frac{50}{41}\mathbf{i}+\frac{40}{41}\mathbf{j}$ and ${{\mathbf{v}}_{\mathbf{1}}}=\frac{50}{41}\mathbf{i}+\frac{40}{41}\mathbf{j}$, ${{\mathbf{v}}_{\mathbf{2}}}=-\frac{132}{41}\mathbf{i}+\frac{165}{41}\mathbf{j}$.
Work Step by Step
The projection vector, $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}$ can be obtained as below:
$\begin{align}
& \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v=}\frac{\mathbf{v}\cdot \mathbf{w}}{{{\left| \mathbf{w} \right|}^{2}}}\mathbf{w} \\
& =\frac{\left( -2\mathbf{i}+5\mathbf{j} \right)\cdot \left( 5\mathbf{i}+4\mathbf{j} \right)}{{{\left( \sqrt{{{5}^{2}}+{{4}^{2}}} \right)}^{2}}}\left( 5\mathbf{i}+4\mathbf{j} \right) \\
& =\frac{-2\left( 5 \right)+5\left( 4 \right)}{{{\left( \sqrt{25+16} \right)}^{2}}}\left( 5\mathbf{i}+4\mathbf{j} \right) \\
& =\frac{-10+20}{{{\left( \sqrt{41} \right)}^{2}}}\left( 5\mathbf{i}+4\mathbf{j} \right)
\end{align}$
We will solve ahead to get the result as,
$\begin{align}
& \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=\frac{10}{41}\left( 5\mathbf{i}+4\mathbf{j} \right) \\
& =\frac{50}{41}\mathbf{i}+\frac{40}{41}\mathbf{j}
\end{align}$
Now, we will obtain ${{\mathbf{v}}_{\mathbf{1}}}$ such that ${{\mathbf{v}}_{\mathbf{1}}}$ is parallel to $\mathbf{w}$ as,
$\begin{align}
& {{\mathbf{v}}_{\mathbf{1}}}\mathbf{=}\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v} \\
& =\frac{50}{41}\mathbf{i}+\frac{40}{41}\mathbf{j}
\end{align}$
Now we will obtain ${{\mathbf{v}}_{\mathbf{2}}}$ such that ${{\mathbf{v}}_{\mathbf{2}}}$ is orthogonal to $\mathbf{w}$ as,
$\begin{align}
& {{\mathbf{v}}_{\mathbf{2}}}=\mathbf{v}-{{\mathbf{v}}_{\mathbf{1}}} \\
& =\left( -2\mathbf{i}+5\mathbf{j} \right)-\left( \frac{50}{41}\mathbf{i}+\frac{40}{41}\mathbf{j} \right) \\
& =-2\mathbf{i}+5\mathbf{j}-\frac{50}{41}\mathbf{i}-\frac{40}{41}\mathbf{j} \\
& =-\frac{132}{41}\mathbf{i}+\frac{165}{41}\mathbf{j}
\end{align}$
Hence, the projection vector, $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=\frac{50}{41}\mathbf{i}+\frac{40}{41}\mathbf{j}$ and ${{\mathbf{v}}_{\mathbf{1}}}=\frac{50}{41}\mathbf{i}+\frac{40}{41}\mathbf{j}$, ${{\mathbf{v}}_{\mathbf{2}}}=-\frac{132}{41}\mathbf{i}+\frac{165}{41}\mathbf{j}$.
