2025 VJC P1 Q3

Referred to the origin \(O\), points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively. Point \(C\) lies on \(OA\), such that \(OC:CA=2:1\). Point \(D\) lies on \(OB\), such that \(OD:DB=\lambda:\mu\). It is given that the area of triangle \(ABD\) is half the area of triangle \(ABC\).

1. Show the area of triangle \(ABD\) is given by \(\frac{\mu}{2(\lambda+\mu)} |\mathbf{a} \times \mathbf{b}|\). Hence find the ratio \(\lambda:\mu\).[4]
2. The point \(E\) has position vector \(\frac{1}{4}\mathbf{a}+\frac{5}{8}\mathbf{b}\). Show that \(A\), \(E\) and \(D\) are collinear.[3]

It is further given that the angle \(AOB\) is \(\frac{\pi}{4}\) and \(O\) lies on the perpendicular bisector of the line segment \(AB\).

3. Find the length of projection of \(\mathbf{a}\) on \(\mathbf{b}\), giving your answer in terms of \(|\mathbf{b}|\). Hence find the position vector of the point \(F\), the foot of perpendicular from \(A\) to \(OB\).[3]