For the lines \(\vec{r} = \vec{a}_1+\lambda\vec{b}_1\) and \(\vec{r} = \vec{a}_2+\mu\vec{b}_2\), it is given that \(\vec{a}_2-\vec{a}_1 = 4\hat{\imath}-\hat{\jmath}+2\hat{k}\) and \(\vec{b}_1\times\vec{b}_2 = 2\hat{\imath}+2\hat{\jmath}-\hat{k}\). The shortest distance between the lines is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{4}{3}\)
The shortest distance between the lines is
\( \displaystyle d = \frac{|(\vec{b}_1\times\vec{b}_2)\cdot(\vec{a}_2-\vec{a}_1)|}{|\vec{b}_1\times\vec{b}_2|} \)
\((2)(4)+(2)(-1)+(-1)(2) = 8-2-2 = 4\)
\(|\vec{b}_1\times\vec{b}_2| = \sqrt{4+4+1} = 3\)
\( \displaystyle d = \frac{4}{3} \)
Why the other options are incorrect
Option (1) 4 — Incorrect. This is only the numerator; it must be divided by \(|\vec{b}_1\times\vec{b}_2| = 3\).
Option (2) \(\displaystyle\frac{4}{9}\) — Incorrect. This divides by \(|\vec{b}_1\times\vec{b}_2|^2 = 9\) instead of 3.
Option (4) \(\displaystyle\frac{4}{\sqrt{21}}\) — Incorrect. This divides by \(|\vec{a}_2-\vec{a}_1| = \sqrt{21}\) instead of \(|\vec{b}_1\times\vec{b}_2|\).