### Instruction: Find the projection of $\mathbf{a}$ onto $\mathbf{b} = \begin{pmatrix} 1 \\ -3 \end{pmatrix}$ if $\mathbf{a} \cdot \mathbf{b} = 2.$.### Response: I know that the projection of $\mathbf{a}$ onto $\mathbf{b}$ is a scalar multiple of $\mathbf{b}$, and that scalar is given by the formula $\frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}}.$
So I need to find the dot product of $\mathbf{b}$ with itself, which is $1^2 + (-3)^2 = 10.$
Then I can plug in the given values into the formula and get $\frac{2}{10} \mathbf{b} = \frac{1}{5} \begin{pmatrix} 1 \\ -3 \end{pmatrix}.$
This is the projection of $\mathbf{a}$ onto $\mathbf{b}$, so I'm done.