The distance between the lines \(\displaystyle\frac{x}{1}=\frac{y}{2}=\frac{z}{2}\) and \(\displaystyle\frac{x-3}{1}=\frac{y-3}{2}=\frac{z}{2}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3
Both lines have direction ratios 1, 2, 2, so they are parallel. The first passes through the origin and the second through (3, 3, 0).
\(\vec{b} = \hat{\imath}+2\hat{\jmath}+2\hat{k},\quad \vec{a}_2-\vec{a}_1 = 3\hat{\imath}+3\hat{\jmath}\)
\(\vec{b}\times(\vec{a}_2-\vec{a}_1) = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ 1 & 2 & 2 \\ 3 & 3 & 0 \end{vmatrix}\)
\(= \hat{\imath}(0-6)-\hat{\jmath}(0-6)+\hat{k}(3-6) = -6\hat{\imath}+6\hat{\jmath}-3\hat{k}\)
\(|-6\hat{\imath}+6\hat{\jmath}-3\hat{k}| = \sqrt{36+36+9} = 9\)
\(|\vec{b}| = \sqrt{1+4+4} = 3\)
\( \displaystyle d = \frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|} = \frac{9}{3} = 3 \)
Why the other options are incorrect
Option (1) 9 — Incorrect. This is \(|\vec{b}\times(\vec{a}_2-\vec{a}_1)|\) before dividing by \(|\vec{b}| = 3\).
Option (2) \(3\sqrt{2}\) — Incorrect. This is the distance from the origin to (3, 3, 0), and that segment is not perpendicular to the lines.
Option (3) 1 — Incorrect. This divides by \(|\vec{b}|^2 = 9\) instead of \(|\vec{b}| = 3\).