2021 ASRJC Promo Q8

Relative to 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 vectors. The point \(C\) is on \(BA\) produced such that \(BA:BC=3:5\) and \(OC\) is perpendicular to \(OB\).

1. Find \(\overrightarrow{OC}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[1]
2. Show that \(\mathbf{a}\cdot \mathbf{b}=\frac{2}{5}{{\left| \mathbf{b} \right|}^{2}}\).[2]

The point \(P\) is on the line \(OB\) such that it is the image of \(B\) in the line \(OC\).

3. Find the area of triangle \(PCB\). Leave your answer in the form of \(k\left| \mathbf{a}\times \mathbf{b} \right|\), where \(k\) is an exact real constant.[3]

The point \(F\) is the foot of perpendicular of \(P\) to the line \(AB\).

4. Given that \({{\left| \mathbf{a} \right|}^{2}}=\frac{7}{25}{{\left| \mathbf{b} \right|}^{2}}\), find the position vector of \(F\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[4]