The angle between two lines whose direction ratio are proportional to \( (-\theta,-2\theta,\theta) \) and \( (-2\theta,\theta,\theta) \), where \( \theta\in\mathrm{R}\sim\{0\} \), is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \cos^{-1}\left(-\frac{1}{6}\right) \)
Since θ ≠ 0, the direction ratios can be divided by θ:
Line 1: \( (-1,\ -2,\ 1) \), line 2: \( (-2,\ 1,\ 1) \)
Two intersecting lines make two angles, α and π − α, which add up to π. Reversing the direction of line 1, its direction ratios are (1, 2, −1). Then
\( \cos\phi = \displaystyle\frac{1(-2)+2(1)+(-1)(1)}{\sqrt6\,\sqrt6} = -\displaystyle\frac{1}{6} \)
So \( \cos^{-1}\left(-\frac16\right) \) is one of the two angles between the lines. The other is \( \cos^{-1}\frac16 \), which is not an option.
Why the other options are incorrect
Option (1) \( \cos^{-1}\left(\frac{1}{3}\right) \) — Incorrect. The cosine of either angle is \( \pm\frac16 \), not \( \frac13 \).
Option (3) \( \displaystyle\frac{\pi}{2} \) — Incorrect. That needs the sum of the products of the direction ratios to be 0, but it is ±1.
Option (4) \( \displaystyle\frac{\pi}{4} \) — Incorrect. That needs \( \cos = \frac{1}{\sqrt2} \), not \( \pm\frac16 \).