At 12 p.m. a hiker X X\,X has position vector (−2i+3j)(-2\mathbf{i} + 3\mathbf{j})(−2i+3j) km relative to a fixed origin O O\,O and moves with constant velocity (4i−2j)(4\mathbf{i} - 2\mathbf{j})(4i−2j) km h−1\text{h}^{-1}h−1. Another hiker Y Y\,Y has position vector (6i−5j)(6\mathbf{i} - 5\mathbf{j})(6i−5j) km relative to a fixed origin O O\,O and moves with constant velocity (−4i+6j)(-4\mathbf{i} + 6\mathbf{j})(−4i+6j) km h−1\text{h}^{-1}h−1.
Find expressions for the position vectors of X X\,X and YYY, in terms of t t\,t hours after 12 p.m.
Show that if both hikers maintain their course and speed, they will collide and find the time and position vector at which this occurs.
At 12:30 p.m. hiker X X\,X changes course and now moves with velocity (10i+10j)(10\mathbf{i} + 10\mathbf{j})(10i+10j) km h−1\text{h}^{-1}h−1. Find the distance between the two hikers at the time when they would have collided.
