2024 NYJC Promo Q9

The plane \(\Pi_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}\), where \(\lambda\) and \(\mu\) are parameters. The plane \(\Pi_2\) has equation \(x - 4y + 5z = 12\).

1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\).[3]
2. Find a vector equation of the line of intersection between \(\Pi_1\) and \(\Pi_2\).[2]
3. The line \(l\) passes through the point \(A\) with position vector \(m\mathbf{i} + (2m + 1)\mathbf{j} - 3\mathbf{k}\) and is parallel to \(3n\mathbf{i} - 3\mathbf{j} + n\mathbf{k}\), where \(m\) and \(n\) are positive constants. Given that the perpendicular distance from \(A\) to the plane \(\Pi_1\) is \(\frac{15}{\sqrt{6}}\) and that the acute angle between \(l\) and \(\Pi_1\) is \(\sin^{-1} \frac{2}{\sqrt{6}}\), find the values of \(m\) and \(n\).[6]