2023 EJC Promo Q12

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

Let the centre of the \(\mathrm{C}\)-atom be the point \(P\), and the centres of the \(4\) \(\mathrm{H}\)-atoms be the points \(Q\), \(R\), \(S\) and \(T\). The coordinates of \(P\), \(Q\), \(R\) and \(S\) are \(\left( -1,\,\,0,\,\,2 \right)\), \(\left( -2,\,\,-1,\,\,1 \right)\), \(\left( -2,\,\,1,\,\,3 \right)\) and \(\left( 0,\,\,-1,\,\,\alpha \right)\) respectively.

The angle \(\theta \) subtended by any two \(\mathrm{C-H}\)bonds at the \(\mathrm{C}\)-atom, such as angle \(QPR\), is known as the \(\mathrm{H-C-H}\) bond angle (see diagram above).

1. Find the bond angle, correct to \(2\) decimal places.[3]
2. By using the fact that \(QS=RS\), show that \(a=3\).[2]
3. Find a cartesian equation for plane \(\pi \), which contains the points \(P\), \(Q\) and \(R\).[3]
4. \(F\) is the point on \(\pi \) that is closest to the point \(S\).
   1. State a vector equation for the line \(SF\).[1]
   2. Hence, show that the coordinates of \(F\) are \(\left( 0,\,\,0,\,\,2 \right)\).[3]
   3. Given that the point \(T\) is the mirror image of the point \(S\) in \(\pi \), find the position vector of \(T\).[2]