### Instruction: Find the reflection of $\begin{pmatrix} 0 \\ 4 \end{pmatrix}$ over the vector $\begin{pmatrix} 1 \\ 3 \end{pmatrix}.$.### Response: To find the reflection of a point over a vector, I need to use the formula $r = 2p \cdot v \frac{v}{\|v\|^2} - p$, where $r$ is the reflection, $p$ is the point, and $v$ is the vector.
I'll plug in the given values of $p$ and $v$ into the formula and simplify.
$r = 2 \begin{pmatrix} 0 \\ 4 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 3 \end{pmatrix} \frac{\begin{pmatrix} 1 \\ 3 \end{pmatrix}}{\| \begin{pmatrix} 1 \\ 3 \end{pmatrix} \|^2} - \begin{pmatrix} 0 \\ 4 \end{pmatrix}$
$= 2 \times 12 \times \frac{1}{10} \begin{pmatrix} 1 \\ 3 \end{pmatrix} - \begin{pmatrix} 0 \\ 4 \end{pmatrix}$
$= \frac{24}{10} \begin{pmatrix} 1 \\ 3 \end{pmatrix} - \begin{pmatrix} 0 \\ 4 \end{pmatrix}$
$= \begin{pmatrix} 2.4 \\ 7.2 \end{pmatrix} - \begin{pmatrix} 0 \\ 4 \end{pmatrix}$