The shortest distance between the x-axis and the line \(\vec{r} = (\hat{\jmath}+3\hat{k})+\mu(2\hat{\imath}+\hat{\jmath})\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
The x-axis is \(\vec{r} = \vec{0}+\lambda\hat{\imath}\). So
\(\vec{a}_1 = \vec{0},\quad \vec{b}_1 = \hat{\imath}\)
\(\vec{a}_2 = \hat{\jmath}+3\hat{k},\quad \vec{b}_2 = 2\hat{\imath}+\hat{\jmath}\)
The shortest distance between the lines is
\( \displaystyle d = \frac{|(\vec{b}_1\times\vec{b}_2)\cdot(\vec{a}_2-\vec{a}_1)|}{|\vec{b}_1\times\vec{b}_2|} \)
\(\vec{b}_1\times\vec{b}_2 = \hat{\imath}\times(2\hat{\imath}+\hat{\jmath}) = \hat{k}\)
\(\hat{k}\cdot(\hat{\jmath}+3\hat{k}) = 3\)
\( \displaystyle d = \frac{3}{|\hat{k}|} = 3 \)
Why the other options are incorrect
Option (1) 1 — Incorrect. 1 is \(|\vec{b}_1\times\vec{b}_2|\), the denominator, not the distance.
Option (3) \(\sqrt{10}\) — Incorrect. This is \(|\vec{a}_2-\vec{a}_1|\), the distance from the origin to (0, 1, 3). That segment is not perpendicular to the second line, so it is not the shortest.
Option (4) 0 — Incorrect. Every point of the line has z = 3, so it never meets the x-axis, and it is not parallel to it.