3. The position vectors \(\mathbf { r } _ { A }\) and \(\mathbf { r } _ { B }\), in kilometres, of two small aeroplanes \(A\) and \(B\) relative to a fixed point \(O\) are given by
$$\begin{aligned}
& \mathbf { r } _ { A } = ( 60 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) + ( 168 \mathbf { i } + 132 \mathbf { j } ) t
& \mathbf { r } _ { B } = ( 62 \mathbf { i } + 3 \mathbf { k } ) + ( 160 \mathbf { i } + p \mathbf { j } + q \mathbf { k } ) t
\end{aligned}$$
where \(t\) denotes the time in hours after 9:00 a.m. and \(p , q\) are constants.
The aeroplanes \(A\) and \(B\) are on course to collide.
- Show that \(p = 140\) and \(q = 4\).
- Find an expression for the square of the distance between \(A\) and \(B\) at time \(t\) hours after 9:00 a.m.
- Both aeroplanes have systems that activate an alarm if they come within 600 m of each other. Determine the time when the alarms are first activated.