2024 ACJC P2 Q2

Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively and they lie on plane \(\pi\) .

1. Show that \(\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}\) is a vector perpendicular to the plane \(\pi\) .[2]
2. Prove that the equation of plane \(\pi\) can be written as
   \[\mathbf{r}\bullet(\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}) = \mathbf{a}\bullet(\mathbf{b}\times\mathbf{c}),\]
   explaining clearly the reason for any result that you use in your proof.[2]
3. Given that \(\mathbf{a} = \mathbf{i}\), \(\mathbf{b} = \mathbf{j}\) and \(\mathbf{c} = \mathbf{k}\), show that the equation of the plane \(\pi\) can be written as \(\mathbf{r}\bullet\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 1\). Hence, find the cartesian equations of the planes which are at a distance of 5 units from plane \(\pi\).[3]