The unit vectors i\mathbf{i}i and j\mathbf{j}j are due east and due north respectively. At midday a boat A A\,A is 5 km east of a fixed origin O O\,O and is moving with constant velocity (−6i+5j)(-6\mathbf{i} + 5\mathbf{j})(−6i+5j) km h−1\text{h}^{-1}h−1. At the same time, another boat B B\,B is 10 km north of O O\,O and is moving with uniform velocity (−4i+j)(-4\mathbf{i} + \mathbf{j})(−4i+j) km h−1\text{h}^{-1}h−1.
Show that, at time t t\,t hours after midday, the position vector of A A\,A is [(5−6t)i+5tj][(5 - 6t)\mathbf{i} + 5t\mathbf{j}][(5−6t)i+5tj] km 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−5)i+(10−4t)j][(2t - 5)\mathbf{i} + (10 - 4t)\mathbf{j}][(2t−5)i+(10−4t)j] km
By using your answer to part (b), or otherwise, show that the boats would collide if they continued at the same velocities and find the time at which the collision would occur.
158 exam-style questions on OCR (MEI) A Level Maths 3.2 Kinematics, covering 3.2.1 Language of kinematics, 3.2.2 Position, displacement, distance and distance travelled, 3.2.3 Velocity, speed and acceleration distinctions, 3.2.4 Draw and interpret kinematics graphs, 3.2.5 Differentiate position and velocity (A-level only), 3.2.6 Integrate acceleration and velocity (A-level only), 3.2.7 When constant acceleration formulae apply, 3.2.8 Solve 1-D kinematics problems, 3.2.9 Language of kinematics in 2 dimensions (A-level only), 3.2.10 Extend 1-D techniques to 2-D using vectors (A-level only), 3.2.11 Cartesian equation of a path (A-level only), and 3.2.12 Vectors to solve kinematics problems (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.