The unit vectors i\mathbf{i}i and j\mathbf{j}j are due east and due north respectively. At 09:00, a drone P P\,P is 4 km east of a fixed origin O O\,O and is moving with constant velocity (−5i+6j)(-5\mathbf{i} + 6\mathbf{j})(−5i+6j) km h−1\text{h}^{-1}h−1. At the same time, another drone Q Q\,Q is 12 km north of O O\,O and is moving with uniform velocity (−4i+3j)(-4\mathbf{i} + 3\mathbf{j})(−4i+3j) km h−1\text{h}^{-1}h−1.
Show that, at time t t\,t hours after 09:00, the position vector of P P\,P is [(4−5t)i+6tj][(4 - 5t)\mathbf{i} + 6t\mathbf{j}][(4−5t)i+6tj] km and find a similar expression for the position vector of Q Q\,Q at this time.
Hence show that, at time ttt, the position vector of Q Q\,Q relative to P P\,P is [(t−4)i+(12−3t)j][(t - 4)\mathbf{i} + (12 - 3t)\mathbf{j}][(t−4)i+(12−3t)j] km
By using your answer to part (b), or otherwise, show that the drones would collide if they continued at the same velocities and find the time at which the collision would occur.