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.
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.