2020 CJC Promo Q5

The diagram shows a circle with centre \(O\) and diameter \(PQ\). The point \(R\) lies on the circumference of the circle. Taking the centre \(O\) as the origin, the position vectors of the points \(P\), \(Q\) and \(R\) are \(\mathbf{p}\), \(\mathbf{q}\) and \(\mathbf{r}\) respectively.

1. By first writing down \(\overrightarrow{PR}\) and \(\overrightarrow{QR}\) in terms of \(\mathbf{p}\) and \(\mathbf{r}\), prove that the lines \(PR\)and \(QR\) are perpendicular, showing your working clearly.[4]

It is given that \(\mathbf{q}\times \mathbf{r}=\left( \mathbf{q}-\mathbf{r} \right)\times \mathbf{s}\) where \(\mathbf{s}\) is the position vector of the point \(S\).

2. By considering \(\overrightarrow{SR}\times \overrightarrow{QR}\), show that \(\overrightarrow{SR}=k\overrightarrow{QR}\), \(k\in \mathbb{R}\), \(k\ne 0\).[3]