Fluency exercise

## Problem

Here are the equations of 12 straight lines.

$\mathbf{r}= \begin{pmatrix} 0 \\ -6 \\ \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 4 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 0 \\ -3 \\ \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 8 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 2 \\ 9.5 \\ \end{pmatrix} + \mu \begin{pmatrix} -2 \\ -8 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 0 \\ 2 \\ \end{pmatrix} + \mu \begin{pmatrix} -2 \\ 1 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 1 \\ 1 \\ \end{pmatrix} + \mu \begin{pmatrix} 4 \\ 1 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 0 \\ 4 \\ \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 4 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 0 \\ -4 \\ \end{pmatrix} + \mu \begin{pmatrix} 4 \\ 6 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 0 \\ 6 \\ \end{pmatrix} + \mu \begin{pmatrix} -2 \\ 10 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} -4 \\ 2 \\ \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -1 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 0 \\ -4 \\ \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 6 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 0 \\ 11 \\ \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -6 \end{pmatrix}$

$\mathbf{r}= \begin{pmatrix} 1 \\ -2 \\ \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 2 \end{pmatrix}$

These 12 straight lines can be divided up into six pairs, each pair matching one of the following descriptions:

• These lines are parallel.

• These lines are perpendicular.

• These lines have the same $y$-intercept.

• These lines have the same $x$-intercept.

• These lines both go through the point $(1, 5)$.

• These lines …

Can you sort them into the correct pairs and complete the final description?