2018 HCI P2 Q4

A line \(l\) has equation \(x-1=\frac{y}{2}=z-3,\) and a plane \(p\) has equation

\(\mathbf{r}=\left( \begin{matrix} 5 \\ -2 \\ 1 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 2 \\ 1 \\ -3 \\ \end{matrix} \right)+\mu \left( \begin{matrix} k \\ 1 \\ 4 \\ \end{matrix} \right),\)

where \(k\) is a real constant, \(\lambda ,\mu \in \mathbb{R}.\)

1. Given that \(l\) and \(p\) are parallel, show that \(k=-1.\)[3]

Use \(k=-1\) for the rest of the parts of the question.

2. Hence show that \(l\) and \(p\) do not intersect.[2]
3. Find the exact distance between \(l\) and \(p.\)[3]
4. A point \(A\) on \(l\) has coordinates \(\left( 2,2,4 \right)\) and \(N\)is the foot of the perpendicular from \(A\) to \(p.\) Find the coordinates of \(N.\) Hence find the coordinates of the reflection of \(N\) in \(l.\)[4]