2010 MJC P2 Q2

Three non-zero and non-parallel vectors \(\mathbf{p}\),\(\mathbf{q}\) and \(\mathbf{r}\) are such that \(\mathbf{p}\times \mathbf{q}=3\mathbf{p}\times \mathbf{r}\). Show that \(\mathbf{q}-3\mathbf{r}=\lambda \mathbf{p}\), where \(\lambda \) is a scalar.[2]

It is also given that \(\mathbf{p}\) is a unit vector, \(\left| \mathbf{q} \right|=5\), \(\left| \mathbf{r} \right|=2\) and the angle between \(\mathbf{q}\) and \(\mathbf{r}\) is \({{\cos }^{-1}}\frac{5}{6}\). By considering \(\left( \mathbf{q}-3\mathbf{r} \right)\cdot \left( \mathbf{q}-3\mathbf{r} \right)\), find the exact values of \(\lambda \).[4]