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.