The distance between the lines \(\vec{r} = (\hat{\imath}+2\hat{\jmath})+\lambda\hat{k}\) and \(\vec{r} = (4\hat{\imath}-2\hat{\jmath}+5\hat{k})+\mu(2\hat{k})\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5
The direction vectors \(\hat{k}\) and \(2\hat{k}\) are parallel, so the lines are parallel. Taking \(\vec{b} = \hat{k}\):
\(\vec{a}_2-\vec{a}_1 = 3\hat{\imath}-4\hat{\jmath}+5\hat{k}\)
\(\hat{k}\times(3\hat{\imath}-4\hat{\jmath}+5\hat{k}) = 4\hat{\imath}+3\hat{\jmath}\)
\(|4\hat{\imath}+3\hat{\jmath}| = \sqrt{16+9} = 5\)
\( \displaystyle d = \frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|} = \frac{5}{1} = 5 \)
This is the distance between the points (1, 2) and (4, −2) in the xy-plane, since both lines are parallel to the z-axis.
Why the other options are incorrect
Option (1) 7 — Incorrect. The distance is \(\sqrt{3^2+4^2} = 5\), not \(3+4 = 7\).
Option (2) \(\sqrt{50}\) — Incorrect. This is \(|\vec{a}_2-\vec{a}_1|\), which also counts the difference of 5 along the z-axis, the direction of the lines.
Option (4) \(\displaystyle\frac{5}{2}\) — Incorrect. This divides by \(|2\hat{k}| = 2\) while the cross product was taken with \(\hat{k}\); the same vector must be used in both places.