The unit vectors i\mathbf{i}i and j\mathbf{j}j are directed east and north. At time t=0t = 0t=0 seconds, runner A A\,A is 10 m east of a fixed marker O O\,O and is running with constant velocity (−4i+5j)(-4\mathbf{i} + 5\mathbf{j})(−4i+5j) m s−1\text{s}^{-1}s−1. At the same time, runner B B\,B is 15 m north of O O\,O and is running with constant velocity (−2i+2j)(-2\mathbf{i} + 2\mathbf{j})(−2i+2j) m s−1\text{s}^{-1}s−1.
Show that, at time t t\,t seconds, the position vector of A A\,A is [(10−4t)i+5tj][(10 - 4t)\mathbf{i} + 5t\mathbf{j}][(10−4t)i+5tj] m and find a similar expression for the position vector of B B\,B at this time.
Hence show that, at time ttt, the position vector of B B\,B relative to A A\,A is [(2t−10)i+(15−3t)j][(2t - 10)\mathbf{i} + (15 - 3t)\mathbf{j}][(2t−10)i+(15−3t)j] m
By using your answer to part (b), or otherwise, show that the runners would collide if they continued at the same velocities and find the time at which the collision would occur.