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.
Practise AQA A Level Maths 1.13 J: Vectors with exam-style questions for A Level Maths. 122 questions covering 1.13.1 Vectors in two and three dimensions, 1.13.2 Magnitude and direction of a vector, 1.13.3 Vector arithmetic, 1.13.4 Position vectors, and 1.13.5 Vectors to solve problems, matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
