The shortest distance between the lines \( \vec{r}=\ 2\hat{\imath}+3\hat{\jmath}+4\hat{k}+\ \lambda(\hat{\imath}-\hat{\jmath}+\hat{k}) \) and \( \vec{r}=5\hat{\imath}+5\hat{\jmath}+5\hat{k}+\mu(-\hat{\imath}+\hat{\jmath}-\hat{k}) \) where \( \lambda,\ \mu\in R \) (set of real numbers) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \sqrt{\displaystyle\frac{38}{3}} \)
The direction \( -\hat{\imath}+\hat{\jmath}-\hat{k} \) is the negative of \( \hat{\imath}-\hat{\jmath}+\hat{k} \), so the lines are parallel. For parallel lines with direction \( \vec{b} \):
\( \displaystyle d=\frac{|(\vec{a}_2-\vec{a}_1)\times\vec{b}|}{|\vec{b}|} \)
\( \displaystyle \vec{a}_2-\vec{a}_1=3\hat{\imath}+2\hat{\jmath}+\hat{k} \)
\( \displaystyle \vec{b}=\hat{\imath}-\hat{\jmath}+\hat{k} \)
\( \displaystyle (\vec{a}_2-\vec{a}_1)\times\vec{b}=\begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\3 & 2 & 1\\1 & -1 & 1\end{vmatrix} \)
\( \displaystyle =3\hat{\imath}-2\hat{\jmath}-5\hat{k} \)
\( \displaystyle d=\frac{\sqrt{9+4+25}}{\sqrt{3}} \)
\( \displaystyle =\frac{\sqrt{38}}{\sqrt{3}}=\sqrt{\frac{38}{3}} \)
Why the other options are incorrect
Option (2) \( \sqrt{38} \) — Incorrect. This is only the numerator; it must be divided by \( |\vec{b}|=\sqrt{3} \).
Option (3) \( 2\sqrt{38} \) — Incorrect. The distance is \( \sqrt{\frac{38}{3}} \), not \( 2\sqrt{38} \).
Option (4) 0 — Incorrect. The distance is 0 only for lines that meet. These lines are parallel and distinct, so the distance is not 0.