2023 TMJC Promo Q8

The plane \(\pi \) contains the point \(A\left( 1,-3,1 \right)\) and the line

\({{l}_{1}}:\mathbf{r}=\left( \begin{matrix} \begin{matrix} 1 \\ -1 \\ 3 \\ \end{matrix} \\ \end{matrix} \right)+\lambda \left( \begin{matrix} \begin{matrix} 2 \\ -1 \\ 1 \\ \end{matrix} \\ \end{matrix} \right)\), \(\lambda \in \mathbb{R}\).

1. Show that the cartesian equation of plane \(\pi \) is \(x+y-z=-3\).[3]
2. Find the position vector of the foot of the perpendicular from the point \(B\left( 0,1,0 \right)\) to the plane \(\pi \).[3]
3. Hence find the position vector of the point \({B}'\) which is the reflected point of \(B\) in the plane \(\pi \).[2]
4. The line \({{l}_{2}}\) is parallel to vector \(\left( \begin{matrix} \begin{matrix} \alpha \\ 1 \\ \beta \\ \end{matrix} \\ \end{matrix} \right)\) where \(\alpha \) and \(\beta \) are real constants. Find the relationship between \(\alpha \) and \(\beta \) such that \({{l}_{2}}\) meets \(\pi \) at a unique point.[2]
5. Given that \(\alpha =4\) and \(\beta =6\), find the acute angle between \({{l}_{2}}\) and \(\pi \).[2]