EJC Vector III Assignment Q1

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
   \(\mathbf{r}\cdot \left( \mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a} \right)=\mathbf{a}\cdot \left( \mathbf{b}\times \mathbf{c} \right)\),
   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 \({\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 1.}\)
   Hence, find the cartesian equations of the planes which are at a distance of 5 units from plane \(\pi \).[3]