7. [In this question the vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors in the direction due east and due north respectively.]
Two boats \(A\) and \(B\) are moving with constant velocities. Boat \(A\) moves with velocity \(9 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Boat \(B\) moves with velocity \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
- Find the bearing on which \(B\) is moving.
At noon, \(A\) is at point \(O\), and \(B\) is 10 km due west of \(O\). At time \(t\) hours after noon, the position vectors of \(A\) and \(B\) relative to \(O\) are \(\mathbf { a } \mathrm { km }\) and \(\mathbf { b } \mathrm { km }\) respectively.
- Find expressions for \(\mathbf { a }\) and \(\mathbf { b }\) in terms of \(t\), giving your answer in the form \(p \mathbf { i } + q \mathbf { j }\).
- Find the time when \(B\) is due south of \(A\).
At time t hours after noon, the distance between \(A\) and \(B\) is \(d \mathrm {~km}\). By finding an expression for \(\overrightarrow { A B }\),
- show that \(d ^ { 2 } = 25 t ^ { 2 } - 60 t + 100\).
At noon, the boats are 10 km apart.
- Find the time after noon at which the boats are again 10 km apart.
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