At 2 p.m. a drone U U\,U has position vector (5i+8j)(5\mathbf{i} + 8\mathbf{j})(5i+8j) km relative to a fixed origin O O\,O and moves with constant velocity (10i−12j)(10\mathbf{i} - 12\mathbf{j})(10i−12j) km h−1\text{h}^{-1}h−1. Another drone V V\,V has position vector (11i−10j)(11\mathbf{i} - 10\mathbf{j})(11i−10j) km relative to a fixed origin O O\,O and moves with constant velocity (−2i+24j)(-2\mathbf{i} + 24\mathbf{j})(−2i+24j) km h−1\text{h}^{-1}h−1.
Find expressions for the position vectors of U U\,U and VVV, in terms of t t\,t hours after 2 p.m.
Show that if both drones maintain their course and speed, they will collide and find the time and position vector at which this occurs.
At 2:15 p.m. drone U U\,U changes course and now moves with velocity (18i−4j)(18\mathbf{i} - 4\mathbf{j})(18i−4j) km h−1\text{h}^{-1}h−1. Find the distance between the two drones at the time when they would have collided.
