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 8 km east of a fixed origin O O\,O and is moving with constant velocity (−6i+4j)(-6\mathbf{i} + 4\mathbf{j})(−6i+4j) km h−1\text{h}^{-1}h−1. At the same time, another boat B B\,B is 12 km north of O O\,O and is moving with uniform velocity (−2i−2j)(-2\mathbf{i} - 2\mathbf{j})(−2i−2j) km h−1\text{h}^{-1}h−1. At time t t\,t hours after midday, the position vector of B B\,B relative to A A\,A is [(4t−8)i+(12−6t)j][(4t - 8)\mathbf{i} + (12 - 6t)\mathbf{j}][(4t−8)i+(12−6t)j] km.
By using the position vector of B B\,B relative to A A\,A given above, 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.
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.