The coordinates of the image of the point P (5, 4, 2) in the line \(\vec{r} = (-\hat{\imath} + 3\hat{\jmath} + \hat{k}) + \lambda(2\hat{\imath} + 3\hat{\jmath} - \hat{k})\), where \(\lambda\) is a parameter, is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (-3, 8, -2)
The foot of the perpendicular Q from P to the line is the midpoint of P and its image \(P'\). A general point of the line is
\(Q(-1 + 2\lambda,\ 3 + 3\lambda,\ 1 - \lambda)\)
\(\vec{PQ} = (-6 + 2\lambda)\hat{\imath} + (-1 + 3\lambda)\hat{\jmath} + (-1 - \lambda)\hat{k}\)
\(\vec{PQ}\) is perpendicular to the direction \(2\hat{\imath} + 3\hat{\jmath} - \hat{k}\) of the line:
\(2(-6 + 2\lambda) + 3(-1 + 3\lambda) - (-1 - \lambda) = 0\)
\(14\lambda - 14 = 0,\quad \lambda = 1\)
So Q = (1, 6, 0). Since Q is the midpoint of \(PP'\):
\(P' = 2Q - P = (2 - 5,\ 12 - 4,\ 0 - 2) = (-3, 8, -2)\)