A laser beam used in a large-scale surveying exercise follows a path represented by the line l l\,l with equation
r=(−215)+μ(21−2) \mathbf{r} = \begin{pmatrix} -2 \\ 1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} r=−215+μ21−2where μ \mu\,μ is a scalar parameter.
Two sensors, P P\,P and QQQ, are located at points on the path lll. Given that
find the value of the constant u u\,u and the value of the constant vvv.
Hence find the vector PQ→\overrightarrow{PQ}PQ.
An auxiliary marker is placed at the point R(−2,2,3)R(-2, 2, 3)R(−2,2,3).
Find the size of angle RPQRPQRPQ, giving your answer in degrees to one decimal place.
A target S S\,S is placed on the line l l\,l such that the area of triangle PRS PRS\,PRS is exactly twice the area of triangle PRQPRQPRQ.
Find the coordinates of the two possible positions of SSS.
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.