### Instruction: Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 2 \\ -11 \end{pmatrix}.\]Find $\text{proj}_{-\mathbf{w}} (\mathbf{v})$.### Response: The problem asks me to find the projection of $\mathbf{v}$ onto the opposite direction of $\mathbf{w}$, which is the same as the negative of the projection of $\mathbf{v}$ onto $\mathbf{w}$.
So, I can use the fact that \[\text{proj}_{-\mathbf{w}} (\mathbf{v}) = -\text{proj}_{\mathbf{w}} \mathbf{v}.\]
I already know that $\text{proj}_{\mathbf{w}} \mathbf{v}$ is given by $\begin{pmatrix} 2 \\ -11 \end{pmatrix}$, so I can just multiply this by -1 to get the answer.
Thus, \[\text{proj}_{-\mathbf{w}} (\mathbf{v}) = -\begin{pmatrix} 2 \\ -11 \end{pmatrix} = \begin{pmatrix} -2 \\ 11 \end{pmatrix}.\]