2025 MI P1 Q11

Referred to the origin \(O\), the 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. The point \(C\) lies on \(OA\) such that \(OC : CA = 1 : 2\). The point \(D\) lies on \(OB\) such that \(OB : DB = 3 : 1\).

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 \(AD\) and \(BC\) meet has position vector \[\frac{1}{7}\mathbf{a} + \frac{4}{7}\mathbf{b}\][5]
3. Show that the area of triangle \(OAE\) can be written as \(k|\mathbf{a} \times \mathbf{b}|\), where \(k\) is a constant to be found.[3]

It is further given that \(\mathbf{b}\) is a unit vector.

4. Give the geometrical meaning of \(|\mathbf{a} \cdot \mathbf{b}|\).[1]
5. If \(|\mathbf{a} \cdot \mathbf{b}| = \frac{1}{2}|\mathbf{a}|\), find the perpendicular distance of point \(A\) to \(OB\), leaving your answer in terms of \(|\mathbf{a}|\).[3]