Here are vector equations of three lines.

\(\mathbf{a}=\begin{pmatrix}4\\0\\-2\end{pmatrix}+\alpha\begin{pmatrix}-1\\1\\4\end{pmatrix}\)

\(\mathbf{b}=\begin{pmatrix}1\\3\\10\end{pmatrix}+\beta\begin{pmatrix}-2\\-1\\2\end{pmatrix}\)

\(\mathbf{c}=\begin{pmatrix}2\\-1\\0\end{pmatrix}+\gamma\begin{pmatrix}1\\2\\2\end{pmatrix}\)

Before doing any calculations,

- try to visualise the lines in 3D space. What can you say about them?
- do you think any of them might be parallel or perpendicular?
- might they intersect?

Now check your predictions and work out what geometrical shape or pattern the lines make.