Relative to a fixed tracking station at the origin OOO, a marine research submersible reaches a first waypoint A A\,A with position vector
OA→=(−245) metres \overrightarrow{OA} = \begin{pmatrix} -2 \\ 4 \\ 5 \end{pmatrix} \text{ metres} OA=−245 metresThe submersible then travels along a displacement vector
AB→=(5−1−2) metres \overrightarrow{AB} = \begin{pmatrix} 5 \\ -1 \\ -2 \end{pmatrix} \text{ metres} AB=5−1−2 metresto reach a second waypoint, BBB.
Determine the coordinates of the waypoint BBB.
An automated sensor is located at a point C C\,C with position vector
OC→=(1k2) metres \overrightarrow{OC} = \begin{pmatrix} 1 \\ k \\ 2 \end{pmatrix} \text{ metres} OC=1k2 metreswhere k k\,k is a scalar constant. Given that the sensor's position vector OC→\overrightarrow{OC}OC is perpendicular to the vector BC→\overrightarrow{BC}BC, representing the path from the second waypoint to the sensor,
find the possible values of kkk.
