Consider the lines \(L_1: \vec{r} = (2\hat{\imath}-\hat{\jmath}+\hat{k})+\lambda(\hat{\imath}+\hat{\jmath}-\hat{k})\) and \(L_2: \vec{r} = (-\hat{\imath}-2\hat{\jmath}-2\hat{k})+\mu(2\hat{\imath}+\hat{\jmath}+\hat{k})\).
(A) \(L_1\) and \(L_2\) are parallel.
(B) \(L_1\) and \(L_2\) intersect at the point (3, 0, 0).
(C) The shortest distance between \(L_1\) and \(L_2\) is 0.
(D) \(L_1\) and \(L_2\) are perpendicular.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B) and (C) only
The direction ratios 1, 1, −1 and 2, 1, 1 are not proportional, so the lines are not parallel.
At a common point, comparing the components:
\(2+\lambda = -1+2\mu\)
\(-1+\lambda = -2+\mu\)
\(1-\lambda = -2+\mu\)
The last two give \(-1+\lambda = 1-\lambda\), so \(\lambda = 1\) and \(\mu = 2\). The first equation also holds: \(3 = 3\).
Point \(= (2+1,\ -1+1,\ 1-1) = (3, 0, 0)\)
For the angle, the dot product of the directions is
\((1)(2)+(1)(1)+(-1)(1) = 2\)
So the lines are not parallel, they meet at (3, 0, 0), and the dot product of their directions is 2.
Checking the statements
(A) \(L_1\) and \(L_2\) are parallel — Incorrect. 2, 1, 1 is not a multiple of 1, 1, −1.
(B) \(L_1\) and \(L_2\) intersect at the point (3, 0, 0) — Correct. \(\lambda = 1\) and \(\mu = 2\) both give (3, 0, 0).
(C) The shortest distance between \(L_1\) and \(L_2\) is 0 — Correct. The lines have the common point (3, 0, 0).
(D) \(L_1\) and \(L_2\) are perpendicular — Incorrect. \(2+1-1 = 2 \ne 0\).