2020 RI P2 Q5

Referred to the origin \(O\), points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively, where \(\mathbf{a}\) and \(\mathbf{b}\) are non-zero and non-parallel vectors. Point \(C\) lies on \(OA\), between \(O\) and \(A\), such that \(OC:CA=2:1\). Point \(D\) lies on \(OB\) produced such that \(OD:BD=3:2\).

1. Find the position vectors \(\overrightarrow{OC}\) and \(\overrightarrow{OD}\), giving your answers in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[2]
2. Show that the point \(E\) where the lines \(BC\) and \(AD\) meet has position vector \(\frac{4}{3}\mathbf{a}-\mathbf{b}\).[4]
3. Show that the area of triangle \(CDE\) can be written as \(k\left| \mathbf{a}\times \mathbf{b} \right|\), where \(k\) is a constant to be found.[3]
4. It is given that the point \(F\) is on \(BO\) produced, and \(OE\) bisects the angle \(AOF\). Find the ration \(OA:OB\).[3]