Vectors: Properties of Parallelism and Intersections

In the diagram, \(OA\) is parallel to \(CB\) and \(SC\) is parallel to \(AB\). The position vectors of \(A\) and \(B\) relative to \(O\) are \(4\mathbf{a}\) and \(5\mathbf{b}\) respectively. It is given that \(OS : SB = 2 : 3\) and \(\overrightarrow{AT} = \frac{1}{3}\overrightarrow{TB}\).

1. Find, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), the vectors
   1. \(\overrightarrow{OS}\),
   2. \(\overrightarrow{AS}\),
   3. \(\overrightarrow{OT}\).
2. \(U\) is a point on \(OT\) such that \(OU : UT = 8 : 3\).
   1. Express \(\overrightarrow{AU}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
   2. Hence write down two facts about \(A\), \(U\) and \(S\).
3. Prove that \(\Delta SBC\) is similar to \(\Delta BOA\).
4. Find
   1. \(\frac{\text{Area of}\hspace{0.5em} \Delta SBC}{\text{Area of}\hspace{0.5em} \Delta BOA}\),
   2. \(\frac{\text{Area of}\hspace{0.5em} \Delta OTA}{\text{Area of}\hspace{0.5em} \Delta OBA}\).