2024 CJC P2 Q1

The points \(P\), \(Q\) and \(R\) have position vectors \(\mathbf{j}+\mathbf{k}\), \(\mathbf{i}-2\mathbf{j}+3\mathbf{k}\) and \(-2\mathbf{i}+\mathbf{j}+4\mathbf{k}\) respectively. Find the exact length of projection of \(\overrightarrow{QR}\) on \(\overrightarrow{PQ}\).[3]

Two non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\) are such that \(\left| \mathbf{a} \right|=\lambda \left| \mathbf{b} \right|\) and \(\left| \mathbf{a}+\lambda \mathbf{b} \right|=2\left| \mathbf{a}-\lambda \mathbf{b} \right|\), where \(\lambda \) is a positive constant. Using a suitable scalar product, show that \(\cos \theta =\frac{3}{5}\), where \(\theta \) is the acute angle between \(\mathbf{a}\) and \(\mathbf{b}\).[3]