The shortest distance between the lines \(\vec{r} = (\hat{\imath} + 2\hat{\jmath} + 3\hat{k}) + \lambda(2\hat{\imath} + 3\hat{\jmath} + 4\hat{k})\) and \(\vec{r} = (2\hat{\imath} + 4\hat{\jmath} + 5\hat{k}) + \mu(4\hat{\imath} + 6\hat{\jmath} + 8\hat{k})\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\sqrt{\displaystyle\frac{5}{29}}\)
The direction vector of the second line is \(4\hat{\imath} + 6\hat{\jmath} + 8\hat{k} = 2(2\hat{\imath} + 3\hat{\jmath} + 4\hat{k})\), so the lines are parallel. Let
\(\vec{b} = 2\hat{\imath} + 3\hat{\jmath} + 4\hat{k}\)
\(\vec{a}_2 - \vec{a}_1 = (2 - 1)\hat{\imath} + (4 - 2)\hat{\jmath} + (5 - 3)\hat{k} = \hat{\imath} + 2\hat{\jmath} + 2\hat{k}\)
The distance between parallel lines is
\(\displaystyle d = \frac{|\vec{b} \times (\vec{a}_2 - \vec{a}_1)|}{|\vec{b}|}\)
\(\displaystyle \vec{b} \times (\vec{a}_2 - \vec{a}_1) = \begin{vmatrix}\hat{\imath}&\hat{\jmath}&\hat{k}\\2&3&4\\1&2&2\end{vmatrix}\)
\(= (6 - 8)\hat{\imath} - (4 - 4)\hat{\jmath} + (4 - 3)\hat{k} = -2\hat{\imath} + \hat{k}\)
\(|\vec{b} \times (\vec{a}_2 - \vec{a}_1)| = \sqrt{4 + 0 + 1} = \sqrt{5}\)
\(|\vec{b}| = \sqrt{4 + 9 + 16} = \sqrt{29}\)
\(\displaystyle d = \frac{\sqrt{5}}{\sqrt{29}} = \sqrt{\frac{5}{29}}\)
Why the other options are incorrect
Option (1) \(\sqrt{5}\) — Incorrect. This is \(|\vec{b} \times (\vec{a}_2 - \vec{a}_1)|\) before dividing by \(|\vec{b}| = \sqrt{29}\).
Option (2) \(\sqrt{29}\) — Incorrect. This is \(|\vec{b}|\), the length of the direction vector, not a distance between the lines.
Option (3) \(\sqrt{\displaystyle\frac{29}{5}}\) — Incorrect. This is the reciprocal of the distance, \(\frac{|\vec{b}|}{|\vec{b} \times (\vec{a}_2 - \vec{a}_1)|}\).