Two automated sensors are located at fixed points P P\,P and Q Q\,Q with position vectors rP=(−142)\mathbf{r}_P = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}rP=−142 and rQ=(226)\mathbf{r}_Q = \begin{pmatrix} 2 \\ 2 \\ 6 \end{pmatrix}rQ=226 respectively, relative to a tracking station at the origin OOO. A surveillance drone flies along a straight line path l1 l_1\,l1 that passes through both P P\,P and QQQ.
Determine an equation for the drone's path l1 l_1\,l1 in the form r=a+λd\mathbf{r} = \mathbf{a} + \lambda \mathbf{d}r=a+λd.
A second drone follows a flight path l2 l_2\,l2 given by the equation
r=(50−2)+μ(113) \mathbf{r} = \begin{pmatrix} 5 \\ 0 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix} r=50−2+μ113Show that the flight paths l1 l_1\,l1 and l2 l_2\,l2 do not intersect.
A beacon is dropped from the second drone at the point C C\,C on l2 l_2\,l2 where μ=−1\mu = -1μ=−1.
Calculate the acute angle between the straight line segment PC PC\,PC and the flight path l2l_2l2. Give your answer in degrees to one decimal place.
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.