7. Two trains \(S\) and \(T\) are moving with constant speeds on straight tracks which intersect at the point \(O\). At 9.00 a.m. \(S\) has position vector \(( - 10 \mathbf { i } + 24 \mathbf { j } ) \mathrm { km }\) and \(T\) has position vector \(25 \mathbf { j }\) km relative to \(O\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due east and due north respectively. \(S\) is moving with speed \(52 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and \(T\) is moving with speed \(50 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), both towards \(O\).

1. Show that the velocity vector of \(S\) is \(( 20 \mathbf { i } - 48 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and find the velocity vector of \(T\).
2. Find expressions for the position vectors of \(S\) and \(T\) at time \(t\) minutes after 9.00 a.m.
3. Show that the bearing of \(T\) from \(S\) remains constant during the motion, and find this bearing.
4. Show that if the trains continue at the given speeds they will collide.