The plane \(\pi\) has equation \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}+ \mu \mathbf{c}\) and the plane \(\pi'\) has equation \(\mathbf{r} = \mathbf{d}+ \lambda'\mathbf{e}+\mu'\mathbf{f}\), where \(\lambda\), \(\mu\), \(\lambda'\) and \(\mu'\) are scalar parameters. It is given that

\[\mathbf{a}=\begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix}, \mathbf{b}=\begin{pmatrix}2\\ -1 \\ -1\end{pmatrix}, \mathbf{c} = \begin{pmatrix}-1 \\ 2 \\ 1\end{pmatrix}, \mathbf{d} = \begin{pmatrix}0 \\ 1 \\ 1\end{pmatrix}, \mathbf{e} = \begin{pmatrix}p \\ q \\ r\end{pmatrix}, \mathbf{f} = \begin{pmatrix}q\\ r\\ p\end{pmatrix},\]

\(\big \vert\mathbf{e}\big \vert=\big \vert\mathbf{f}\big \vert = \sqrt{30}\) and that \(p\) is negative. The planes \(\pi\) and \(\pi'\) are parallel. Find \(p\), \(q\) and \(r\).

Show that \(\pi'\) is the reflection of \(\pi\) in the origin.

The line \(L\) has equation \[\dfrac{x+1}{9}=\dfrac{y-1}{-6}=\dfrac{z}{-5}\] and lies in \(\pi\). The point \(P\) is in \(\pi\) with parameters \(\lambda = l\) and \(\mu = m\), and \(P'\) is in \(\pi'\) with \(\lambda' = l\) and \(\mu' = m\). The point \(Q'\) is the reflection of \(P\) in the origin. Show that when \(P\) lies on \(L\) the direction of \(P'Q'\) is independent of the position of \(P\) on \(L\).