1 A mine contains several underground tunnels beneath a hillside. The hillside is a plane, all the tunnels are straight and the width of the tunnels may be neglected. A coordinate system is chosen with the \(z\)-axis pointing vertically upwards and the units are metres. Three points on the hillside have coordinates \(\mathrm { A } ( 15 , - 60,20 )\), \(B ( - 75,100,40 )\) and \(C ( 18,138,35.6 )\).

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\) and hence show that the equation of the hillside is \(2 x - 2 y + 25 z = 650\). The tunnel \(T _ { \mathrm { A } }\) begins at A and goes in the direction of the vector \(15 \mathbf { i } + 14 \mathbf { j } - 2 \mathbf { k }\); the tunnel \(T _ { \mathrm { C } }\) begins at C and goes in the direction of the vector \(8 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\). Both these tunnels extend a long way into the ground.
2. Find the least possible length of a tunnel which connects B to a point in \(T _ { \mathrm { A } }\).
3. Find the least possible length of a tunnel which connects a point in \(T _ { \mathrm { A } }\) to a point in \(T _ { \mathrm { C } }\).
4. A tunnel starts at B , passes through the point ( \(18,138 , p\) ) vertically below C , and intersects \(T _ { \mathrm { A } }\) at the point Q . Find the value of \(p\) and the coordinates of Q .