### Instruction: Find the reflection of $\begin{pmatrix} 0 \\ 4 \end{pmatrix}$ over the vector $\begin{pmatrix} 1 \\ 3 \end{pmatrix}.$.### Response: To reflect a point over a vector, I need to find the projection of the point onto the vector, and then double the displacement from the point to the projection.
The projection of $\begin{pmatrix} 0 \\ 4 \end{pmatrix}$ onto $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$ is given by the formula $\dfrac{\begin{pmatrix} 0 \\ 4 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 3 \end{pmatrix}}{\begin{pmatrix} 1 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 3 \end{pmatrix}} \begin{pmatrix} 1 \\ 3 \end{pmatrix}$, where $\cdot$ denotes the dot product.
Computing the dot products, I get $\dfrac{12}{10} \begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 1.2 \\ 3.6 \end{pmatrix}$.
The displacement from $\begin{pmatrix} 0 \\ 4 \end{pmatrix}$ to $\begin{pmatrix} 1.2 \\ 3.6 \end{pmatrix}$ is $\begin{pmatrix} 1.2 \\ -0.4 \end{pmatrix}$.
Doubling this displacement gives $\begin{pmatrix} 2.4 \\ -0.8 \end{pmatrix}$.
Adding this to the original point $\begin{pmatrix} 0 \\ 4 \end{pmatrix}$ gives the reflection $\begin{pmatrix} 2.4 \\ 3.2 \end{pmatrix}$.