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+μ113
Show 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 Edexcel A Level Maths Vectors with exam-style questions for A Level Maths. 100 questions covering 12.1 3D Coordinates, 12.2 Vectors in 3D, 12.3 Solving Geometric Problems, and 12.4 Application to Mechanics, matched to the Edexcel A Level Maths (9MA0) 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.