2024 DHS P1 Q8

The line \({{l}_{1}}\) passes through the point \(A\left( 1,\ 2,\ 4 \right)\) and point \(B\left( -1,\ -1,\ 3 \right)\). The line \({{l}_{2}}\) has equation

\(\mathbf{r}=\left( \begin{matrix} \begin{matrix} 3 \\ -5 \\ 2 \end{matrix} \end{matrix} \right)+t\left( \begin{matrix} \begin{matrix} 6 \\ -7 \\ 2 \end{matrix} \end{matrix} \right)\), \(t\in \mathbb{R}\).

1. Explain why \({{l}_{1}}\) and \({{l}_{2}}\) are skew lines.[2]
2. Find an equation of the plane, in scalar product form, that includes the midpoint of \(AB\) and the line \({{l}_{2}}\).[3]
3. Find the coordinates of \(C\).[3]

The plane \({{\pi }_{2}}\) has equation \(3x-4y+\lambda z=\mu \) and line \({{l}_{1}}\) does not intersect \({{\pi }_{2}}\).

4. Find the value of \(\lambda \). Hence find the acute angle between the planes \({{\pi }_{1}}\) and \({{\pi }_{2}}\).[3]
5. If the distance between \({{\pi }_{2}}\) and \({{l}_{1}}\) is 2 units, find the exact values of \(\mu \).[3]