At 6 a.m. a boat A A\,A has position vector (12i−11j)(12\mathbf{i} - 11\mathbf{j})(12i−11j) km relative to a fixed origin O O\,O and moves with constant velocity (9i−6j)(9\mathbf{i} - 6\mathbf{j})(9i−6j) km h−1\text{h}^{-1}h−1. Another boat B B\,B has position vector (40i−39j)(40\mathbf{i} - 39\mathbf{j})(40i−39j) km relative to a fixed origin O O\,O and moves with constant velocity (−12i+15j)(-12\mathbf{i} + 15\mathbf{j})(−12i+15j) km h−1\text{h}^{-1}h−1.
Find expressions for the position vectors of A A\,A and BBB, in terms of t t\,t hours after 6 a.m.
Show that if both boats maintain their course and speed, they will collide and find the time and position vector at which this occurs.
At 7 a.m. boat A A\,A realises that a collision is imminent and changes course so that it now has velocity (−18i+21j)(-18\mathbf{i} + 21\mathbf{j})(−18i+21j) km h−1\text{h}^{-1}h−1. Find the distance between the two ships at the time when they would have collided.
197 exam-style questions on OCR A Level Maths 1.10 Vectors, covering 1.10.1 Vectors in two dimensions, 1.10.2 Vectors in three dimensions (A-level only), 1.10.3 Magnitude and direction of vectors, 1.10.4 Basic operations on vectors, 1.10.5 Position vectors, 1.10.6 Distance between points, 1.10.7 Problem solving using vectors, 1.10.8 Vectors in kinematics, and 1.10 Vectors. Each one has a worked solution and a mark scheme showing where the marks go.