At 6 a.m. a boat A A\,A has position vector (10i−8j)(10\mathbf{i} - 8\mathbf{j})(10i−8j) km relative to a fixed origin O O\,O and moves with constant velocity (6i−3j)(6\mathbf{i} - 3\mathbf{j})(6i−3j) km h−1\text{h}^{-1}h−1. Another boat B B\,B has position vector (30i−28j)(30\mathbf{i} - 28\mathbf{j})(30i−28j) km relative to a fixed origin O O\,O and moves with constant velocity (−9i+12j)(-9\mathbf{i} + 12\mathbf{j})(−9i+12j) 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 (−12i+15j)(-12\mathbf{i} + 15\mathbf{j})(−12i+15j) km h−1\text{h}^{-1}h−1. Find the distance between the two ships at the time when they would have collided.
363 exam-style questions on Edexcel A Level Maths Further Kinematics, covering 8.1 Vectors in Kinematics, 8.2 Vector Methods with Projectiles, 8.3 Variable Acceleration in One Dimension, 8.4 Differentiating Vectors, and 8.5 Integrating Vectors. Each one has a worked solution and a mark scheme showing where the marks go.