2024 CJC P2 Q4

The plane \(p\) has equation \(\mathbf{r}=\left( \begin{matrix} \begin{matrix} 1 \\ 3 \\ 4 \\ \end{matrix} \\ \end{matrix} \right)+s\left( \begin{matrix} \begin{matrix} 2 \\ 0 \\ 1 \\ \end{matrix} \\ \end{matrix} \right)+t\left( \begin{matrix} \begin{matrix} 0 \\ 1 \\ -1 \\ \end{matrix} \\ \end{matrix} \right)\), and the line \(l\) has the equation \(\mathbf{r}=\left( \begin{matrix} \begin{matrix} 1 \\ 1 \\ 3 \\ \end{matrix} \\ \end{matrix} \right)+\lambda \left( \begin{matrix} \begin{matrix} -1 \\ 2 \\ 2 \\ \end{matrix} \\ \end{matrix} \right)\), where \(\lambda \), \(s\) and \(t\) are parameters.

1. Show that \(l\) is perpendicular to \(p\) and find the coordinates of the point where \(l\) and \(\mathrm{p}\) intersect.[5]

The line \(m\) meets \(l\) at the point \(A\left( 1,1,3 \right)\) and it meets \(p\) at the point \(B\left( 1,3,4 \right)\).

2. The line \(n\) is the reflection of \(m\) in \(p\). Show that a vector equation of \(n\) is \(\mathbf{r}=\left( \begin{matrix} \begin{matrix} 1 \\ 3 \\ 4 \\ \end{matrix} \\ \end{matrix} \right)+\mu \left( \begin{matrix} \begin{matrix} -4 \\ 2 \\ 5 \\ \end{matrix} \\ \end{matrix} \right)\), \(\mu \in \mathbb{R}\).[2]
3. Find the acute angle between \(n\) and \(p\).[2]
4. Hence or otherwise, find the area bounded by the lines \(l\), \(m\) and \(n\).[3]