To solve the problem, we need to find the angle between two lines whose direction cosines \( l, m, n \) satisfy the equations \( l + m + n = 0 \) and \( lm = 0 \).
### Step 1: Analyze the given equations
We have two equations:
1. \( l + m + n = 0 \) (Equation 1)
2. \( lm = 0 \) (Equation 2)
From Equation 2, since \( lm = 0 \), either \( l = 0 \) or \( m = 0 \).
### Step 2: Case 1 - \( l = 0 \)
If \( l = 0 \), substitute into Equation 1:
\[
0 + m + n = 0 \implies m + n = 0 \implies m = -n
\]
Let’s denote \( n = k \). Then we have:
\[
m = -k
\]
Thus, the direction ratios become:
\[
(0, -k, k) \quad \text{or} \quad (0, -1, 1) \quad \text{(taking \( k = 1 \))}
\]
### Step 3: Case 2 - \( m = 0 \)
If \( m = 0 \), substitute into Equation 1:
\[
l + 0 + n = 0 \implies l + n = 0 \implies l = -n
\]
Let’s denote \( n = k \). Then we have:
\[
l = -k
\]
Thus, the direction ratios become:
\[
(-k, 0, k) \quad \text{or} \quad (-1, 0, 1) \quad \text{(taking \( k = 1 \))}
\]
### Step 4: Calculate the angle between the lines
We have two sets of direction ratios:
1. From Case 1: \( (0, -1, 1) \)
2. From Case 2: \( (-1, 0, 1) \)
Using the formula for the cosine of the angle \( \theta \) between two lines with direction ratios \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \):
\[
\cos \theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}}
\]
Substituting the values from Case 1 and Case 2:
\[
\cos \theta = \frac{|0 \cdot (-1) + (-1) \cdot 0 + 1 \cdot 1|}{\sqrt{0^2 + (-1)^2 + 1^2} \cdot \sqrt{(-1)^2 + 0^2 + 1^2}}
\]
Calculating the numerator:
\[
= \frac{|0 + 0 + 1|}{\sqrt{0 + 1 + 1} \cdot \sqrt{1 + 0 + 1}} = \frac{1}{\sqrt{2} \cdot \sqrt{2}} = \frac{1}{2}
\]
### Step 5: Find the angle \( \theta \)
Now, we know:
\[
\cos \theta = \frac{1}{2}
\]
This implies:
\[
\theta = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3}
\]
### Final Answer
Thus, the angle between the two lines is \( \frac{\pi}{3} \).
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