Two high-precision calibration lasers, Beam AAA and Beam BBB, are used to align a medical imaging scanner. Relative to a fixed origin OOO at the scanner's base, the paths of the beams are modeled by the following equations:
Beam A:rA=(3−11)+s(14−2) \text{Beam } A: \mathbf{r}_A = \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix} Beam A:rA=3−11+s14−2 Beam B:rB=(110)+t(−3−21) \text{Beam } B: \mathbf{r}_B = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + t \begin{pmatrix} -3 \\ -2 \\ 1 \end{pmatrix} Beam B:rB=110+t−3−21where sss and ttt are scalar parameters. The beams are known to intersect at the system's isocenter, XXX.
Find the position vector of XXX.
A diagnostic sensor is placed at the point P(6,11,−5)P(6, 11, -5)P(6,11,−5) on Beam AAA. A target point QQQ is located on Beam BBB.
Given that the vector PQ⃗\vec{PQ}PQ is perpendicular to the path of Beam BBB, determine the exact coordinates of QQQ.
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.