2015 CJC P1 Q2

With reference to the origin \(O\), two points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively. The points \(O\), \(A\) and \(B\) are not collinear. The point \(P\) divides \(AB\) in the ratio \(AP:PB=3:2\) . It is given that \(\mathbf{a}\) is a unit vector, \(OB=4\), angle \(AOB=\frac{\pi }{3}\) and the foot of the perpendicular from \(P\) to the line passing through points \(O\) and \(A\) is \(F\) . Show that \(\overrightarrow{OF}=\lambda \mathbf{a}\) , where \(\lambda\) is a constant to be determined.[5]