For $\bold{v} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}$ and $\bold{w} = \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}$, compute $\text{proj}_{\bold{w}} \bold{v}$.
We have that
\begin{align*}
\text{proj}_{\bold{w}} \bold{v} &= \frac{\bold{v} \cdot \bold{w}}{\bold{w} \cdot \bold{w}} \bold{w} \\
&= \frac{\begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}}{\begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}} \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix} \\
&= \frac{1}{5} \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix} \\
&= \boxed{\begin{pmatrix} 2/5 \\ -1/5 \\ 0 \end{pmatrix}}.
\end{align*}