At 10 a.m. an aircraft P P\,P has position vector (−5i+10j)(-5\mathbf{i} + 10\mathbf{j})(−5i+10j) km relative to a fixed origin O O\,O and moves with constant velocity (200i+150j)(200\mathbf{i} + 150\mathbf{j})(200i+150j) km h−1\text{h}^{-1}h−1. Another aircraft Q Q\,Q has position vector (15i+20j)(15\mathbf{i} + 20\mathbf{j})(15i+20j) km relative to a fixed origin O O\,O and moves with constant velocity (100i+100j)(100\mathbf{i} + 100\mathbf{j})(100i+100j) km h−1\text{h}^{-1}h−1.
Find expressions for the position vectors of P P\,P and QQQ, in terms of t t\,t hours after 10 a.m.
Show that if both aircraft maintain their course and speed, they will collide and find the time and position vector at which this occurs.
At 10:06 a.m. aircraft P P\,P realizes that a collision is imminent and changes course so that it now has velocity (250i+100j)(250\mathbf{i} + 100\mathbf{j})(250i+100j) km h−1\text{h}^{-1}h−1. Find the distance between the two aircraft at the time when they would have collided.
