### Instruction: Find the reflection of $\begin{pmatrix} 0 \\ 4 \end{pmatrix}$ over the vector $\begin{pmatrix} 1 \\ 3 \end{pmatrix}.$.### Response: To reflect a vector over another vector, I need to decompose the vector into a component parallel to the second vector and a component perpendicular to it.
Then, the reflection will have the same parallel component, but the opposite perpendicular component.
To find the parallel component, I can use the dot product formula: $\text{proj}_v(u) = \frac{u \cdot v}{v \cdot v}v,$ where $u$ is the vector to be projected and $v$ is the vector to project onto.
Plugging in the given vectors, I get: $\text{proj}_{\begin{pmatrix} 1 \\ 3 \end{pmatrix}}(\begin{pmatrix} 0 \\ 4 \end{pmatrix}) = \frac{\begin{pmatrix} 0 \\ 4 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 3 \end{pmatrix}}{\begin{pmatrix} 1 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 3 \end{pmatrix}}\begin{pmatrix} 1 \\ 3 \end{pmatrix} = \frac{12}{10}\begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 1.2 \\ 3.6 \end{pmatrix}.$
To find the perpendicular component, I can subtract the parallel component from the original vector: $\begin{pmatrix} 0 \\ 4 \end{pmatrix} - \begin{pmatrix} 1.2 \\ 3.6 \end{pmatrix} = \begin{pmatrix} -1.2 \\ 0.4 \end{pmatrix}.$
The reflection will have the same parallel component, but the opposite perpendicular component, so I can add the parallel component to the negative of the perpendicular component: $\begin{pmatrix} 1.2 \\ 3.6 \end{pmatrix} - \begin{pmatrix} -1.2 \\ 0.4 \end{pmatrix} = \begin{pmatrix} 2.4 \\ 3.2 \end{pmatrix}.$