Question

The direction cosines \(l, m, n\) of two lines are connected by the relations \(l + m + n = 0\) and \(lm = 0\), then the angle between them is

• A. \(\frac{\pi}{3}\)
• B. \(\frac{\pi}{4}\)
• C. \(\frac{\pi}{2}\)
• D. 0

Hint:

For direction cosines, remember \(l^2 + m^2 + n^2 = 1\). Solve system of equations.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Direction cosines satisfy \(l^2 + m^2 + n^2 = 1\). Given \(l + m + n = 0\) and \(lm = 0\).

Step 2: Detailed Explanation:
Since \(lm = 0\), either \(l = 0\) or \(m = 0\).
Case 1: \(l = 0\). Then \(m + n = 0 \implies n = -m\).
Also \(m^2 + n^2 = 1 \implies m^2 + m^2 = 1 \implies 2m^2 = 1 \implies m = \pm \frac{1}{\sqrt{2}}\), \(n = \mp \frac{1}{\sqrt{2}}\).
So direction cosines: \((0, \frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}})\).
Case 2: \(m = 0\). Then \(l + n = 0 \implies n = -l\).
\(l^2 + n^2 = 1 \implies 2l^2 = 1 \implies l = \pm \frac{1}{\sqrt{2}}\), \(n = \mp \frac{1}{\sqrt{2}}\).
So direction cosines: \((\frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}})\).
Angle between them: dot product = \(0 \times \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \times 0 + (-\frac{1}{\sqrt{2}}) \times (-\frac{1}{\sqrt{2}}) = \frac{1}{2}\).
So \(\cos \theta = \frac{1}{2} \implies \theta = \frac{\pi}{3}\).

Step 3: Final Answer:
Option (A) \(\frac{\pi}{3}\).