At 6 a.m. a boat A A\,A has position vector (12i−11j)(12\mathbf{i} - 11\mathbf{j})(12i−11j) km relative to a fixed origin O O\,O and moves with constant velocity (9i−6j)(9\mathbf{i} - 6\mathbf{j})(9i−6j) km h−1^{-1}−1.
At the same time a boat B B\,B has position vector (40i−39j)(40\mathbf{i} - 39\mathbf{j})(40i−39j) km relative to O O\,O and moves with constant velocity (−12i+15j)(-12\mathbf{i} + 15\mathbf{j})(−12i+15j) km h−1^{-1}−1.
Show that, if both boats maintain their course and speed, they will collide, and find the time and the position vector at which this occurs.
At 7 a.m. boat A A\,A recognises the danger and changes course, so that from that moment it has velocity (−18i+21j)(-18\mathbf{i} + 21\mathbf{j})(−18i+21j) km h−1^{-1}−1. Find the distance between the two boats at the time at which they would have collided.