2023 CJC P1 Q3

The points \(P\) , \(Q\) and \(R\) have position vectors \(\mathbf{p}\), \(\mathbf{q}\) and \(\mathbf{r}\) respectively. The points \(P\) and \(Q\) are fixed and \(R\) varies.

1. Given that \(\mathbf{p}\) is non-zero and \(\left( \mathbf{r}-\mathbf{q} \right)\times \mathbf{p}=\mathbf{0}\), find a linear relationship between \(\mathbf{p}\), \(\mathbf{q}\) and \(\mathbf{r}\).
   Describe geometrically the set of all possible positions of the point \(R\).[4]
2. Given that \(\mathbf{r}=\left( \begin{matrix} x \\ y \\ z \\ \end{matrix} \right)\), \(\mathbf{p}=\left( \begin{matrix} 2 \\ -5 \\ 3 \\ \end{matrix} \right)\), \(\mathbf{q}=\left( \begin{matrix} -4 \\ 1 \\ 2 \\ \end{matrix} \right)\) and \(\left( \mathbf{r}-\mathbf{q} \right)\cdot \mathbf{p}=\mathrm{0}\), find a relationship between \(x\), \(y\) and \(z\).
   Describe geometrically the set of all possible positions of the point \(R\).[4]