2023 EJC Promo Q4

Referred to the origin \(O\), let \(P\), \(Q\) and \(R\) be distinct points with position vectors \(\mathbf{p}\), \(\mathbf{q}\) and \(\mathbf{r}\) respectively.

1. Show that \(\left( \mathbf{r}-\mathbf{p} \right)\times \left( \mathbf{r}-\mathbf{q} \right)=\mathbf{p}\times \mathbf{q}+\mathbf{q}\times \mathbf{r}+\mathbf{r}\times \mathbf{p}\).[2]
2. Give the geometrical meaning of \(\frac{1}{2}\left| \mathbf{p}\times \mathbf{q}+\mathbf{q}\times \mathbf{r}+\mathbf{r}\times \mathbf{p} \right|\).[2]
3. Given that \(\mathbf{p}\times \mathbf{q}+\mathbf{q}\times \mathbf{r}+\mathbf{r}\times \mathbf{p}=\mathbf{0}\), \(PR=3QR\), and that \(PQ>PR\), express \(\mathbf{r}\) in terms of \(\mathbf{p}\) and \(\mathbf{q}\).[3]