2019 HCI Promo Q5

The diagram shows a circle centred at \(O\) with radius \(1\) unit.

It is given that \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\) and \(\overrightarrow{OC}=-\frac{2}{3}\mathbf{a}-\frac{1}{3}\mathbf{b}\).

1. Show that the area of triangle \(ABC\) is given by \(\left| \mathbf{a}\times \mathbf{b} \right|\) square units. Given that \(\angle ACB=\frac{\pi }{3}\), find the exact area of triangle \(ABC\).[5]
2. The point \(M\) lies on \(CB\) produced such that the area of triangle \(AMC\) is three times that of the area of triangle \(ABC\). Find \(\overrightarrow{OM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[2]