The unit vectors i\mathbf{i}i and j\mathbf{j}j are due east and due north respectively.
At midday a boat A A\,A is 5 km east of a fixed origin O O\,O and is moving with constant velocity (−6i+5j)(-6\mathbf{i} + 5\mathbf{j})(−6i+5j) km h−1^{-1}−1. At the same time another boat B B\,B is 10 km north of O O\,O and is moving with constant velocity (−4i+j)(-4\mathbf{i} + \mathbf{j})(−4i+j) km h−1^{-1}−1.
Show that, at time t t\,t hours after midday, the position vector of A A\,A is [(5−6t)i+5tj]\left[(5 - 6t)\mathbf{i} + 5t\mathbf{j}\right][(5−6t)i+5tj] km, and find a similar expression for the position vector of BBB.
Hence show that, at time ttt, the position vector of B B\,B relative to A A\,A is [(2t−5)i+(10−4t)j]\left[(2t - 5)\mathbf{i} + (10 - 4t)\mathbf{j}\right][(2t−5)i+(10−4t)j] km.
Using your answer to part (b), show that the boats would collide if they maintained these velocities, and find the time at which the collision would occur.