Find the projection of the vector $\begin{pmatrix} 3 \\ 0 \end{pmatrix}$ onto the vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}.$
From the formula for a projection,
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 0 \end{pmatrix}} \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \frac{\begin{pmatrix} 3 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -3 \end{pmatrix}}{\left\| \begin{pmatrix} 2 \\ -3 \end{pmatrix} \right\|^2} \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \frac{6}{13} \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \boxed{\begin{pmatrix} 12/13 \\ -18/13 \end{pmatrix}}.\]