2019 EJC P1 Q11

Methane \(\left( \mathrm{C}{{\mathrm{H}}_{4}} \right)\) is an example of a chemical compound with a tetrahedral structure. The \(4\) hydrogen \(\mathrm{(H)}\) atoms form a regular tetrahedron, and the carbon \(\left( \mathrm{C} \right)\) atom is in the centre.

Let the \(4\) \(\mathrm{H}\)- atoms be at points \(P\), \(Q\), \(R\), and \(S\) with coordinates \((9,2,9)\), \((9,8,3)\), \((3,2,3)\), and \((3,8,9)\) respectively.

1. Find a Cartesian equation of the plane \(\prod\nolimits_{1}{{}}\)which contains the points \(P\), \(Q\) and \(R\).[4]
2. Find a Cartesian equation of the plane \(\prod\nolimits_{2}{{}}\)which passes through the midpoint of \(PQ\) and is perpendicular to \(\overrightarrow{PQ}\).[2]
3. Find the coordinates of point \(F\), the foot of the perpendicular from \(S\) to \(\prod{_{_{1}}}\).[4]
4. Let \(T\) be the point representing the carbon \(\mathrm{(C)}\) atom. Given that point \(T\) is perpendicular from the points \(P\), \(Q\), \(R\), and \(S\), find the coordinates of \(T\).[3]