The unit vectors i\mathbf{i}i and j\mathbf{j}j are due east and due north respectively. At noon, two survey vessels A A\,A and B B\,B move with constant velocities.
Relative to a fixed origin OOO, A A\,A has position vector 2i−j2\mathbf{i} - \mathbf{j}2i−j km and velocity 4i+j4\mathbf{i} + \mathbf{j}4i+j km h−1^{-1}−1. Vessel B B\,B has position vector 12i+5j12\mathbf{i} + 5\mathbf{j}12i+5j km and velocity i−3j\mathbf{i} - 3\mathbf{j}i−3j km h−1^{-1}−1.
Find the position vector of B B\,B relative to A A\,A at time t t\,t hours after noon.
Determine the time at which the vessels are closest together.
Find their minimum separation.
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.