Direction cosines of 2 lines are given by the equations \( 3l+m+5n=0,6mn-2nl+5lm=0 \) then which of the following statements are correct?
(A) Direction ratios of two lines are (1, 2, -1) and (-2, 1, 1).
(B) Angle between lines is \( \displaystyle\frac{\pi}{3} \).
(C) The angle between the lines is \( \cos^{-1}\left(-\displaystyle\frac{1}{6}\right) \).
(D) Vectors parallel to these lines are \( \hat{\imath}+2\hat{\jmath}-\hat{k} \) and \( -2\hat{\imath}+\hat{\jmath}+\hat{k} \).
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
From the first equation, \( m = -3l-5n \). Substituting in the second:
\( 6n(-3l-5n) - 2nl + 5l(-3l-5n) = 0 \)
\( -15l^2 - 45ln - 30n^2 = 0 \)
\( l^2 + 3ln + 2n^2 = 0 \)
\( (l+n)(l+2n) = 0 \)
l = −n: \( m = 3n-5n = -2n \), direction ratios \( (-1,-2,1) \), that is, \( (1, 2, -1) \)
l = −2n: \( m = 6n-5n = n \), direction ratios \( (-2, 1, 1) \)
For these direction ratios,
\( \cos\theta = \displaystyle\frac{1(-2)+2(1)+(-1)(1)}{\sqrt6\,\sqrt6} = -\displaystyle\frac{1}{6} \)
Two intersecting lines make two angles, θ and π − θ. With these direction ratios the angle is \( \cos^{-1}\left(-\frac16\right) \); the other angle is \( \cos^{-1}\frac16 \).
(A) Direction ratios (1, 2, −1) and (−2, 1, 1) — Correct. As found above.
(B) Angle \( \displaystyle\frac{\pi}{3} \) — Incorrect. That needs \( \cos = \frac12 \), but \( \cos = \pm\frac16 \).
(C) Angle \( \cos^{-1}\left(-\frac16\right) \) — Correct. As found above.
(D) Vectors \( \hat\imath+2\hat\jmath-\hat k \) and \( -2\hat\imath+\hat\jmath+\hat k \) — Correct. Their components are the direction ratios in (A).
NCERT Exemplar:
Three Dimensional Geometry
