Question

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0 $ and $ l^2= m^2 + n^2 $is

• A. $\pi/6$
• B. $\pi/2$
• C. $\pi/3$
• D. $\pi/4$

Answer 1

The Correct Option is C

To find the angle between the lines whose direction cosines satisfy the equations \(l + m + n = 0\) and \( l^2 = m^2 + n^2 \), follow the steps below:

1. Given the direction cosine conditions:

   • l + m + n = 0 (Equation 1)
   • l^2 = m^2 + n^2 (Equation 2)

2. From Equation 1, express \(l\) as:

   l = -(m + n)

3. Substitute \(l = -(m + n)\) into Equation 2:

   (-(m + n))^2 = m^2 + n^2

   This simplifies to:

   m^2 + 2mn + n^2 = m^2 + n^2

4. Cancel out the common terms:

   2mn = 0

   This implies:

   mn = 0

5. The implication \(mn = 0\) suggests either:

   • m = 0, then l = -n or
   • n = 0, then l = -m

6. Based on these conditions, the two possible sets of direction cosines are:

   • (l, m, n) = (0, 1/\sqrt{2}, -1/\sqrt{2})
   • (l, m, n) = (1/\sqrt{2}, 0, -1/\sqrt{2})

7. The angle \theta between the two lines can be found using the formula:

   \cos(\theta) = l_1 l_2 + m_1 m_2 + n_1 n_2

   Substitute the values:

   \cos(\theta) = 0 \cdot \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \cdot 0 + (-\frac{1}{\sqrt{2}})(-\frac{1}{\sqrt{2}})

   Calculate the result:

   \cos(\theta) = 0 + 0 + \frac{1}{2} = \frac{1}{2}

8. Finding the angle, we know:

   \theta = \cos^{-1}(\frac{1}{2}) = \frac{\pi}{3}

   Therefore, the angle between the lines is \frac{\pi}{3}.

Hence, the correct answer is \frac{\pi}{3}.