2020 NJC Promo Q8

Points \(A\), \(B\) and \(C\) are non-collinear and origin \(O\) is not on the plane \(ABC\). Point \(P\) has position vector given by \(\overrightarrow{OP}=\lambda \overrightarrow{OA}+\mu \overrightarrow{OB}+\left( 1-\lambda -\mu \right)\overrightarrow{OC}\), where \(\lambda \), \(\mu \in \mathbb{R}\).

1. Prove that \(P\) lies in the plane \(ABC\).[2]

It is now given that \(\overrightarrow{OA}=\mathbf{i}+\mathbf{j}+\mathbf{k}\), \(\overrightarrow{OB}=2\mathbf{i}+2\mathbf{j}+4\mathbf{k}\) and \(\overrightarrow{OC}=-2\mathbf{i}+\mathbf{k}\). Furthermore, \(P\) is the point of reflection of \(B\) in the line passing through \(A\) and \(C\).

2. Find the value of \(\lambda \) and of \(\mu \).[5]