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 5 km east of a fixed origin O O\,O and is moving with constant velocity (−5i+6j)(-5\mathbf{i} + 6\mathbf{j})(−5i+6j) km h−1\text{h}^{-1}h−1. At the same time, another boat B B\,B is 10 km north of O O\,O and is moving with uniform velocity (pi+qj)(p\mathbf{i} + q\mathbf{j})(pi+qj) km h−1\text{h}^{-1}h−1, where p p\,p and q q\,q are constants. At 14:30, the boats collide.
Show that, at time t t\,t hours after midday, the position vector of A A\,A is [(5−5t)i+6tj][(5 - 5t)\mathbf{i} + 6t\mathbf{j}][(5−5t)i+6tj] km and find a similar expression for the position vector of B B\,B at this time.
Hence show that, at time ttt, the position vector of B B\,B relative to A A\,A is [{(p+5)t−5}i+{10+(q−6)t}j][\{(p + 5)t - 5\}\mathbf{i} + \{10 + (q - 6)t\}\mathbf{j}][{(p+5)t−5}i+{10+(q−6)t}j] km
By using your answer to part (b), or otherwise, find the velocity of BBB.
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.