2022 ASRJC Promo Q9

The line \({{l}_{1}}\) passes through the points \(A\) and \(B\) with position vectors \(\left( \begin{matrix} 1 \\ 2 \\ -1 \\ \end{matrix} \right)\) and \(\left( \begin{matrix} 2 \\ -1 \\ 3 \\ \end{matrix} \right)\) respectively. The line \({{l}_{2}}\) has equation \(\mathbf{r}=\left( \begin{matrix} 1 \\ 2 \\ -1 \\ \end{matrix} \right)+\mu \left( \begin{matrix} 1 \\ 0 \\ -3 \\ \end{matrix} \right)\), \(\mu \in \mathbb{R}\) and passes through the point \(C\). It is given that \(BC\) is perpendicular to line \({{l}_{2}}\).

1. Find the coordinates of \(C\).[4]
2. The line \({{l}_{3}}\) is a reflection of the line \({{l}_{1}}\) in the line \({{l}_{2}}\) . Find a vector equation of the line \({{l}_{3}}\).[3]
3. The point \(P\) has position vector \(6\mathbf{j}+\alpha \,\mathbf{k}\), where \(\alpha \in \mathbb{R}\). The length of projection of \(\overrightarrow{AP}\) onto the \({{l}_{2}}\) is given by \(\sqrt{\frac{5}{2}}\). Find the possible values of \(\alpha \).[2]