2023 EJC P1 Q3

With reference to the origin \(O\), the points \(A\), \(B\) and \(C\) are such that \(\overrightarrow{OA}=\left( \begin{matrix} 2 \\ 1 \\ -1 \\ \end{matrix} \right)\), \(\overrightarrow{OB}=\left( \begin{matrix} 0 \\ 1 \\ -2 \\ \end{matrix} \right)\) and \(\overrightarrow{OC}=\left( \begin{matrix} 1 \\ 3 \\ 0 \\ \end{matrix} \right)\).

Point \(P\) lies on \(AB\), between \(A\) and \(B\), such that \(AP:PB=\lambda :1-\lambda \) where \(0<\lambda <1\). Point \(Q\) lies on \(BC\), between \(B\) and \(C\), such that \(BQ:QC=\lambda :1-\lambda \).

1. Express \(\overrightarrow{PQ}\) in terms of \(\lambda \).[3]

The point \(X\) has coordinates \(\left( 2,\,-1,\,-\frac{5}{2} \right)\). Points \(P\), \(Q\) and \(X\) are collinear.

2. Find the value of \(\lambda \).[3]