Let \(L_1\) and \(L_2\) be two lines, represented as,
\(L_1: \vec{r} = \hat{\imath} + \hat{\jmath} + \lambda(2\hat{\imath} - \hat{\jmath} + \hat{k})\) and \(L_2: \vec{r} = 2\hat{\imath} + \hat{\jmath} - \hat{k} + \mu(4\hat{\imath} - 2\hat{\jmath} + 2\hat{k})\), where \(\lambda\) and \(\mu\) are scalars. Then which of the following are true?
(A) \(L_1\) is perpendicular to \(L_2\)
(B) \(L_1\) is parallel to \(L_2\)
(C) \(L_1\) passes through the point (1, 1, 0)
(D) \(L_2\) passes through the point (2, 1, −1)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (C) and (D) only
The direction vectors are the vectors multiplied by \(\lambda\) and \(\mu\):
\( \vec{b}_1 = 2\hat{\imath} - \hat{\jmath} + \hat{k} \)
\( \vec{b}_2 = 4\hat{\imath} - 2\hat{\jmath} + 2\hat{k} = 2\vec{b}_1 \)
\( \vec{b}_1\cdot\vec{b}_2 = 8 + 2 + 2 = 12 \)
Putting \(\lambda = 0\) in \(L_1\) and \(\mu = 0\) in \(L_2\) gives a point on each line:
\( L_1: (1, 1, 0) \)
\( L_2: (2, 1, -1) \)
So \(\vec{b}_2 = 2\vec{b}_1\), \(\vec{b}_1\cdot\vec{b}_2 = 12\), \(L_1\) passes through (1, 1, 0) and \(L_2\) passes through (2, 1, −1).
Checking the statements
(A) \(L_1\) is perpendicular to \(L_2\) — Incorrect. \(\vec{b}_1\cdot\vec{b}_2 = 12 \ne 0\).
(B) \(L_1\) is parallel to \(L_2\) — Correct. \(\vec{b}_2 = 2\vec{b}_1\), so the directions are proportional.
(C) \(L_1\) passes through the point (1, 1, 0) — Correct. \(\lambda = 0\) gives (1, 1, 0).
(D) \(L_2\) passes through the point (2, 1, −1) — Correct. \(\mu = 0\) gives (2, 1, −1).