Consider the lines \(L_1: \vec{r} = 2\hat{\imath}+\lambda\hat{k}\) and \(L_2: \vec{r} = 3\hat{\jmath}+\mu\hat{\imath}\).
(A) \(L_1\) and \(L_2\) are perpendicular
(B) \(L_1\) and \(L_2\) intersect
(C) The shortest distance between \(L_1\) and \(L_2\) is 3
(D) The point (5, 3, 0) lies on \(L_2\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
The points of the two lines are
\(L_1: (2, 0, \lambda),\qquad L_2: (\mu, 3, 0)\)
The directions are \(\vec{b}_1 = \hat{k}\) and \(\vec{b}_2 = \hat{\imath}\):
\(\vec{b}_1\cdot\vec{b}_2 = \hat{k}\cdot\hat{\imath} = 0\)
Every point of \(L_1\) has y = 0 and every point of \(L_2\) has y = 3, so the lines never meet.
For the shortest distance, \(\vec{a}_2-\vec{a}_1 = -2\hat{\imath}+3\hat{\jmath}\) and
\(\vec{b}_1\times\vec{b}_2 = \hat{k}\times\hat{\imath} = \hat{\jmath}\)
\( \displaystyle d = \frac{|(-2\hat{\imath}+3\hat{\jmath})\cdot\hat{\jmath}|}{|\hat{\jmath}|} = 3 \)
So the lines are perpendicular, they do not meet, the shortest distance is 3, and \(\mu = 5\) gives the point (5, 3, 0) of \(L_2\).
Checking the statements
(A) \(L_1\) and \(L_2\) are perpendicular — Correct. \(\hat{k}\cdot\hat{\imath} = 0\).
(B) \(L_1\) and \(L_2\) intersect — Incorrect. y is 0 on \(L_1\) and 3 on \(L_2\), so they are skew.
(C) The shortest distance between \(L_1\) and \(L_2\) is 3 — Correct. \(d = |(-2\hat{\imath}+3\hat{\jmath})\cdot\hat{\jmath}| = 3\).
(D) The point (5, 3, 0) lies on \(L_2\) — Correct. \(\mu = 5\) gives (5, 3, 0).