Question

The direction cosines of two lines are connected by the relations \( 1 + m - n = 0 \) and \( lm - 2mn + nl = 0 \). If \( \theta \) is the acute angle between those lines, then \( \cos \theta = \) ? 

• A. \( \frac{\pi}{6} \)
• B. \( \frac{1}{\sqrt{7}} \)
• C. \( \frac{5}{6} \)
• D. \( \frac{\sqrt{5}}{6} \)

Hint:

For problems involving direction cosines, express variables in terms of one another using given constraints, and apply dot product formulas to find the angle between two lines.

Answer 1

The Correct Option is B

Step 1: Expressing the Direction Cosines Equations 
The given equations relating the direction cosines \( l, m, n \) are: \[ 1 + m - n = 0 \] \[ lm - 2mn + nl = 0. \] From the first equation: \[ n = 1 + m. \] Substituting \( n = 1 + m \) into the second equation: \[ lm - 2m(1 + m) + (1 + m)l = 0. \] Expanding: \[ lm - 2m - 2m^2 + l + lm = 0. \] Rearranging: \[ 2lm - 2m - 2m^2 + l = 0. \] 
Step 2: Finding the Cosine of the Angle Between the Lines 
Using the dot product formula: \[ \cos \theta = \frac{|l_1 l_2 + m_1 m_2 + n_1 n_2|}{\sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2}}. \] By solving the values of \( l, m, n \) from the given equations and substituting into the cosine formula, we obtain: \[ \cos \theta = \frac{1}{\sqrt{7}}. \] 
Final Answer: \( \boxed{\frac{1}{\sqrt{7}}} \)