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.
260 exam-style questions on OCR A Level Maths 3.2 Kinematics, covering 3.2.1 Language of kinematics, 3.2.2 Graphs in kinematics, 3.2.3 Displacement-time and velocity-time graphs, 3.2.4 Constant acceleration formulae, 3.2.5 Constant acceleration in two dimensions (A-level only), 3.2.6 Non-uniform acceleration in one dimension (A-level only), 3.2.7 Non-uniform acceleration in two dimensions (A-level only), 3.2.8 Motion under gravity using vectors (A-level only), and 3.2.9 Projectiles (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.