The distance between the parallel lines \(\vec{r} = \vec{a}_1+\lambda\vec{b}\) and \(\vec{r} = \vec{a}_2+\mu\vec{b}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|}\)
Let S and T be the points with position vectors \(\vec{a}_1\) and \(\vec{a}_2\), and let P be the foot of the perpendicular from T to the first line. The distance between the lines is TP.
If \(\theta\) is the angle between \(\overrightarrow{ST}\) and \(\vec{b}\), then
\(TP = ST\sin\theta\)
\(|\vec{b}\times\overrightarrow{ST}| = |\vec{b}|\,ST\sin\theta = |\vec{b}|\,TP\)
With \(\overrightarrow{ST} = \vec{a}_2-\vec{a}_1\), the distance between parallel lines is
\( \displaystyle d = \frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{|\vec{b}\cdot(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|}\) — Incorrect. This is \(ST\cos\theta\), the length of SP along the line, not the perpendicular distance.
Option (3) \(|\vec{a}_2-\vec{a}_1|\) — Incorrect. This is ST, the distance between two particular points, which is more than TP unless ST is perpendicular to the lines.
Option (4) \(\displaystyle\frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{a}_2-\vec{a}_1|}\) — Incorrect. This equals \(|\vec{b}|\sin\theta\), which depends on the length of \(\vec{b}\) and is not a distance between the lines.