A laser beam used in a large-scale surveying exercise follows a path represented by the line $l\,$ with equation

\mathbf{r} = \begin{pmatrix} -2 \\ 1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}

where $\mu\,$ is a scalar parameter.

Two sensors, $P\,$ and $Q$ , are located at points on the path $l$ . Given that

find the value of the constant $u\,$ and the value of the constant $v$ .

Hence find the vector $PQ→\overrightarrow{PQ}$ .

An auxiliary marker is placed at the point $R (- 2, 2, 3)$ .

Find the size of angle $R P Q$ , giving your answer in degrees to one decimal place.

A target $S\,$ is placed on the line $l\,$ such that the area of triangle $PRS\,$ is exactly twice the area of triangle $P R Q$ .

Find the coordinates of the two possible positions of $S$ .