Relative to the origin \(O\), the fixed points \(A\) and \(B\) have position vectors \(\mathbf a\) and \(\mathbf b\) respectively. It is given that \(\mathbf a\) and \(\mathbf b\) are non-zero and non-parallel vectors.
The variable point \(R\) has position vector \(\mathbf r\). Given that \((\mathbf r-\mathbf a)\times(\mathbf a-\mathbf b)=\mathbf0\), describe geometrically the set of all possible positions of \(R\).[2]
The variable point \(P\) has position vector \(\mathbf p\). Given that \((\mathbf p-\mathbf a)\cdot(\mathbf a\times\mathbf b)=0\), describe geometrically the set of all possible positions of \(P\).[3]
The variable point \(Q\) with position vector \(\mathbf q\) lies on a plane \(\Pi\) with the equation \(\mathbf q\cdot(\mathbf a\times\mathbf b)=|\mathbf a\times\mathbf b|\). Describe geometrically the relationship between \(\Pi\) and the set of all possible positions of \(P\) found in part (b).[2]