A marine research vessel monitors the positions of underwater sensors relative to a coastal lighthouse located at a fixed origin OOO.
Sensor P P\,P is located at position vector
OP→=(413) km \overrightarrow{OP} = \begin{pmatrix} 4 \\ 1 \\ 3 \end{pmatrix}\text{ km} OP=413 kmSensor Q Q\,Q is deployed such that its displacement from sensor P P\,P is given by
PQ→=(35−4) km \overrightarrow{PQ} = \begin{pmatrix} 3 \\ 5 \\ -4 \end{pmatrix}\text{ km} PQ=35−4 kmDetermine the coordinates of sensor QQQ.
A surface buoy R R\,R is located at position vector
OR→=(1k2) km \overrightarrow{OR} = \begin{pmatrix} 1 \\ k \\ 2 \end{pmatrix}\text{ km} OR=1k2 kmwhere k k\,k is a constant.
Given that the position vector of the buoy, OR→\overrightarrow{OR}OR, is perpendicular to the displacement vector between sensor Q Q\,Q and the buoy, QR→\overrightarrow{QR}QR,
find the possible values of kkk.
