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 (−5i+4j)(-5\mathbf{i} + 4\mathbf{j})(−5i+4j) km h−1\text{h}^{-1}h−1. At the same time, another boat B B\,B is 15 km north of O O\,O and is moving with uniform velocity (−2i+j)(-2\mathbf{i} + \mathbf{j})(−2i+j) km h−1\text{h}^{-1}h−1. At time t t\,t hours after midday, the position vector of A A\,A is [(8−5t)i+4tj][(8 - 5t)\mathbf{i} + 4t\mathbf{j}][(8−5t)i+4tj] km and the position vector of B B\,B is [−2ti+(15+t)j][-2t\mathbf{i} + (15 + t)\mathbf{j}][−2ti+(15+t)j] km.
Hence show that, at time ttt, the position vector of B B\,B relative to A A\,A is [(3t−8)i+(15−3t)j][(3t - 8)\mathbf{i} + (15 - 3t)\mathbf{j}][(3t−8)i+(15−3t)j] km
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.