N2025 P2 Q2

The points \(P\), \(Q\) and \(R\) are collinear. \(P\) and \(Q\) have position vectors \(\begin{pmatrix} -2 \\ 3 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ 6 \\ 6 \end{pmatrix}\) respectively. The point \(R\) is such that \(3\overrightarrow{QR} = 4\overrightarrow{PQ}\).

1. Find the position vector of \(R\).[3]

The point \(S\) has position vector \(\begin{pmatrix} -1 \\ -2 \\ c \end{pmatrix}\), where \(c\) is a constant and \(\overrightarrow{SQ}\) is perpendicular to \(\overrightarrow{PQ}\).

2. Find the value of \(c\).[2]

3. Use a scalar product to find the angle between \(\overrightarrow{PS}\) and \(\overrightarrow{PQ}\).

   [3]