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−3−21
where 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.
Practise Edexcel A Level Maths Vectors with exam-style questions for A Level Maths. 100 questions covering 12.1 3D Coordinates, 12.2 Vectors in 3D, 12.3 Solving Geometric Problems, and 12.4 Application to Mechanics, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.