Two drones, U U\,U and VVV, are operating in an open three-dimensional test range.
Drone U U\,U is initially at rest at a point with position vector (4i−2j+7k)(4\mathbf{i} - 2\mathbf{j} + 7\mathbf{k})(4i−2j+7k) metres. It moves with a constant acceleration of (2i−4j+6k) m s−2(2\mathbf{i} - 4\mathbf{j} + 6\mathbf{k})\, \text{m}\,\text{s}^{-2}(2i−4j+6k)ms−2.
Drone V V\,V moves along a straight line path that passes through the points with position vectors (−1i+3j+2k)(-1\mathbf{i} + 3\mathbf{j} + 2\mathbf{k})(−1i+3j+2k) metres and (5i+dj+20k)(5\mathbf{i} + d\mathbf{j} + 20\mathbf{k})(5i+dj+20k) metres.
The drones are moving along parallel paths.
Show that d=−9d = -9d=−9.
Find an expression for the position vector of drone U U\,U at time t t\,t seconds.
Hence, prove that the paths of drones U U\,U and V V\,V are not collinear.
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.