2019 EJC P2 Q3

The position vectors of points \(P\) and \(Q\), with respect to the origin \(O\), are \(\mathbf{p}\) and \(\mathbf{q}\) respectively. Point \(R\), with position vector \(\mathbf{r}\), is on \(PQ\) produced, such that \(3\overrightarrow{PR}=5\overrightarrow{PQ}\).

1. Given that \(\left| \mathbf{p} \right|=\sqrt{29}\) and \(\mathbf{p}\mathbf{.r}=11\), find the length of projection of \(\overrightarrow{OQ}\) onto \(\overrightarrow{OP}\).[4]
2. \(S\) is on another point such that \(\overrightarrow{PS}=\mathbf{r}\). Given that \(\mathbf{p}=3\mathbf{i}+2\mathbf{j}-4\mathbf{k}\) and \(\mathbf{r}=\mathbf{i}-2\mathbf{j}-3\mathbf{k}\), find the area of the quadrilateral \(OPSR\).[3]