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.
