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+μ21−2
where 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.
