2021 TMJC P2 Q3

With reference to the origin \(O\), the points \(A\) and \(B\) are such that \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\). Point \(D\) is the mid-point of \(OB\) and point \(C\) lies on \(OA\) such that \(2OC=3CA\). The lines \(AD\) and \(BC\) intersect at point \(E\) (see Figure 1).

1. Show that the vector equation of the line \(BC\) can be written as \(\mathbf{r}=\frac{3}{5}\lambda \mathbf{a}+\left( 1-\lambda \right)\mathbf{b}\), where \(\lambda \) is a real parameter.[1]
2. Find the position vector of \(E\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[4]
3. Point \(F\) is such that \(\overrightarrow{OF}\) is in the same direction as \(\overrightarrow{AB}\). Given that the area of trapezium \(OABF\) is \(\frac{13}{16}\left| \mathbf{a}\times \mathbf{b} \right|\) units\(^{2}\), find the position vector of \(F\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[3]