2020 HCI Promo Q5

Relative to the origin \(O\), the position vectors of points \(A\), \(B\) and \(C\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. It is given that \(\left| \mathbf{a} \right|=\sqrt{3}\) and \(\left| \mathbf{c} \right|=4\), and the angle between \(\mathbf{a}\) and \(\mathbf{c}\) is \(60{}^\circ \)

Given that \(\mathbf{a}\times 2\mathbf{b}=3\mathbf{a}\times \mathbf{c}\), and \(3\mathbf{b}\times \mathbf{c}=\mathbf{c}\times 4\mathbf{a}\), show that

\(\left| \overrightarrow{AB}\times \overrightarrow{AC} \right|=k\left| \mathbf{a}\times \mathbf{c} \right|\)

where \(k\) is a constant to be determined.[4]

Hence find the exact area of triangle \(ABC\).[2]