9 The points \(P ( 3,5 , - 21 )\) and \(Q ( - 1,3 , - 16 )\) are on the ceiling of a long straight underground tunnel. A ventilation shaft must be dug from the point \(M\) on the ceiling of the tunnel midway between \(P\) and \(Q\) to horizontal ground level (where the \(z\)-coordinate is 0 ). The ventilation shaft must be perpendicular to the tunnel. The path of the ventilation shaft is modelled by the vector equation \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) is the position vector of \(M\). You are given that \(\mathbf { b } = \left( \begin{array} { l } 1
\mathrm {~s}
\mathrm { t } \end{array} \right)\) where \(s\) and \(t\) are real numbers.

1. Show that \(\mathrm { S } = 2.5 \mathrm { t } - 2\).
2. Show that at the point where the ventilation shaft reaches the ground \(\lambda = \frac { \mathrm { C } } { \mathrm { t } }\), where \(c\) is a constant to be determined.
3. Using the results in parts (a) and (b), determine the shortest possible length of the ventilation shaft.
4. Explain what the fact that \(\mathbf { b } \times \left( \begin{array} { l } 0
   0
   1 \end{array} \right) \neq \mathbf { O }\) means about the direction of the ventilation shaft.