2020 VJC P1 Q7

With reference to the origin \(O\), the points \(A\), \(B\), \(P\), \(Q\) and \(R\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{a}-2\mathbf{b}\), \(-2\mathbf{a}-3\mathbf{b}\) and \(2\mathbf{a}+\mathbf{b}\) respectively, where \(\mathbf{a}\) and \(\mathbf{b}\) are non-parallel vectors and \(\mathbf{a}\cdot \mathbf{b}<0\).

Given that \(\mathbf{a}\) is a unit vector and the area of triangle \(PQR\) is equal to the magnitude of \(\mathbf{b}\), show that \(\sin \theta =\frac{1}{4}\), where \(\theta \) is the angle between \(\mathbf{a}\) and \(\mathbf{b}\).

Hence, find the value of \(\theta \).[5]

\(M\) lies on \(PR\) such that \(PM=\frac{1}{2}MR\). Given that \(PR\) is perpendicular to \(OM\), find the magnitude of \(\mathbf{b}\), giving your answer correct to \(3\) decimal places.[5]