6. [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due East and North respectively.] A coastguard station \(O\) monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship \(A\) is at the point with position vector \(( - 5 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to \(O\) and has velocity \(( 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Ship \(B\) is at the point with position vector \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }\) and has velocity \(( - 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Given that the two ships maintain these velocities, show that they collide. The coast guard radios ship \(A\) and orders it to reduce its speed to move with velocity \(( \mathbf { i } + \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Given that \(A\) obeys this order and maintains this new constant velocity,
2. find an expression for the vector \(\overrightarrow { A B }\) at time \(t\) hours after noon.
3. find, to 3 significant figures, the distance between \(A\) and \(B\) at 1400 hours,
4. find the time at which \(B\) will be due north of \(A\).