Two lines \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) are given as \( \mathrm{L}_1:\vec{r}=(\hat{\imath}+2\hat{\jmath}+3\hat{k})+\lambda(2\hat{\imath}+3\hat{\jmath}+4\hat{k}) \) and \( \mathrm{L}_2:\vec{r}=(2\hat{\imath}+4\hat{\jmath}+5\hat{k})+\mu(4\hat{\imath}+6\hat{\jmath}+8\hat{k}) \), where \( \lambda \) and \( \mu \) are real parameters. Then which of the following statements are correct ?
A. Shortest distance between line \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) is \( \sqrt{\displaystyle\frac{5}{29}} \).
B. line \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) are parallel.
C. line \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) are concurrent lines.
D. line \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) are perpendicular.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A and B only
Here \( \vec{a}_1=\hat{\imath}+2\hat{\jmath}+3\hat{k} \), \( \vec{b}=2\hat{\imath}+3\hat{\jmath}+4\hat{k} \), \( \vec{a}_2=2\hat{\imath}+4\hat{\jmath}+5\hat{k} \), and the direction of \( \mathrm{L}_2 \) is \( 4\hat{\imath}+6\hat{\jmath}+8\hat{k}=2\vec{b} \), so the lines are parallel.
For parallel lines the shortest distance is
\( \displaystyle d=\frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|} \)
\( \displaystyle \vec{a}_2-\vec{a}_1=\hat{\imath}+2\hat{\jmath}+2\hat{k} \)
\( \displaystyle \vec{b}\times(\vec{a}_2-\vec{a}_1)=\begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\2 & 3 & 4\\1 & 2 & 2\end{vmatrix} \)
\( \displaystyle =\hat{\imath}(6-8)-\hat{\jmath}(4-4)+\hat{k}(4-3) \)
\( \displaystyle =-2\hat{\imath}+\hat{k} \)
\( \displaystyle d=\frac{\sqrt{4+0+1}}{\sqrt{4+9+16}} \)
\( \displaystyle =\frac{\sqrt{5}}{\sqrt{29}} \)
\( \displaystyle =\sqrt{\frac{5}{29}} \)
So the lines are parallel (direction vectors in the ratio 1 : 2) and the distance between them is \( \sqrt{\frac{5}{29}} \).
Checking the statements
(A) Shortest distance between line \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) is \( \sqrt{\frac{5}{29}} \). — Correct. The distance is \( \sqrt{\frac{5}{29}} \).
(B) line \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) are parallel. — Correct. The direction vectors are in the ratio 1 : 2.
(C) line \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) are concurrent lines. — Incorrect. Parallel lines at a non-zero distance never meet.
(D) line \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) are perpendicular. — Incorrect. Parallel lines make an angle 0; also \( \vec{b}\cdot 2\vec{b}=58\ne 0 \).
Tip: Check the direction vectors first: if they are proportional, use the parallel-lines formula \( \frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|} \), not the skew-lines formula (whose denominator \( |\vec{b}_1\times\vec{b}_2| \) would be zero).