2022 YIJC P1 Q4

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

1. Given that \(\mathbf{r}\times \mathbf{q}=-\mathbf{q}\times \mathbf{p}\), describe geometrically the set of all possible positions of the point \(R\).[4]
2. Given instead that \(\mathbf{r}=\left( \begin{matrix} x \\ y \\ z \\ \end{matrix} \right)\), \(\mathbf{p}=\left( \begin{matrix} 1 \\ -2 \\ 3 \\ \end{matrix} \right)\), \(\mathbf{q}=\left( \begin{matrix} {{q}_{1}} \\ {{q}_{2}} \\ {{q}_{3}} \\ \end{matrix} \right)\), and that \(\mathbf{q}\cdot \left( \mathbf{p}-\mathbf{r} \right)=0\), find the relationship between \(x\), \(y\) and \(z\) in terms of \({{q}_{1}}\), \({{q}_{2}}\) and \({{q}_{3}}\). Describe the set of all possible positions of the point \(R\) in this case.[3]
3. It is now given that \(\left| \mathbf{q} \right|=1\) and \(C\) is a point with position vector \(\mathbf{c}\) such that \(\mathbf{q}\cdot \left( \mathbf{p}-\mathbf{c} \right)\ne 0\). Give a geometrical meaning of \(\left| \mathbf{q}\cdot \left( \mathbf{p}-\mathbf{c} \right) \right|\).[1]