7 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2
7 - x & - 6 & 1
5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units.
\(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .
- In the case where \(x = 5\)
- find the volume of \(H ^ { \prime }\),
- determine whether A preserves the orientation of \(H\).
- Find the values of \(x\) for which \(O , P ^ { \prime } , Q ^ { \prime }\) and \(R ^ { \prime }\) are coplanar (i.e. the four points lie in the same plane).