The shortest distance between the line through the points A(1, −1, 0) and B(2, 1, 2) and the line through the points C(4, −1, 3) and D(6, 0, 1) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
The directions of the lines are
\(\vec{b}_1 = \overrightarrow{AB} = \hat{\imath}+2\hat{\jmath}+2\hat{k}\)
\(\vec{b}_2 = \overrightarrow{CD} = 2\hat{\imath}+\hat{\jmath}-2\hat{k}\)
They are not parallel, and
\(\vec{b}_1\times\vec{b}_2 = (-4-2)\hat{\imath}-(-2-4)\hat{\jmath}+(1-4)\hat{k} = -6\hat{\imath}+6\hat{\jmath}-3\hat{k}\)
\(|\vec{b}_1\times\vec{b}_2| = \sqrt{36+36+9} = 9\)
With A and C as the given points, \(\overrightarrow{AC} = 3\hat{\imath}+0\hat{\jmath}+3\hat{k}\). The shortest distance is
\( \displaystyle d = \frac{|\overrightarrow{AC}\cdot(\vec{b}_1\times\vec{b}_2)|}{|\vec{b}_1\times\vec{b}_2|} = \frac{|-18+0-9|}{9} = \frac{27}{9} = 3 \)
Why the other options are incorrect
Option (1) 1 — Incorrect. The formula gives \(\frac{27}{9} = 3\), not 1.
Option (3) \(3\sqrt{2}\) — Incorrect. This is the distance AC between two chosen points, \(\sqrt{9+0+9}\); it is not the shortest distance between the lines.
Option (4) 9 — Incorrect. This divides 27 by 3 instead of by \(|\vec{b}_1\times\vec{b}_2| = 9\).