N2025 P1 Q8

The plane \(\pi_1\) has equation \(2x + y + 3z = 2\). A point \(P\) has position vector \(\begin{pmatrix} -3 \\ 1 \\ 0 \end{pmatrix}\).

1. Find the foot of the perpendicular from \(P\) to the plane \(\pi_1\).[4]
2. Find the position vector of the reflection of \(P\) in the plane \(\pi_1\).[2]

A second plane \(\pi_2\) has equation \(\mathbf{r} = \begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix}\), where \(\lambda\) and \(\mu\) are parameters.

3. Find the acute angle between \(\pi_1\) and \(\pi_2\).[3]