To find the angle between the two lines given their direction ratios, we can follow these steps:
### Step 1: Write down the equations for the direction ratios
The direction ratios of the two lines are given by the equations:
1. \( a + b + c = 0 \) (Equation 1)
2. \( 2ab + 2ac - bc = 0 \) (Equation 2)
### Step 2: Simplify Equation 2
From Equation 2, we can factor out \( 2a \):
\[
2a(b + c) - bc = 0
\]
This can be rearranged to:
\[
2a(b + c) = bc
\]
### Step 3: Express \( b + c \) in terms of \( a \)
From Equation 1, we can express \( b + c \):
\[
b + c = -a \quad \text{(Equation 3)}
\]
### Step 4: Substitute Equation 3 into Equation 2
Substituting Equation 3 into the modified Equation 2 gives:
\[
2a(-a) = bc
\]
This simplifies to:
\[
-2a^2 = bc \quad \text{(Equation 4)}
\]
### Step 5: Solve for \( b \) and \( c \)
From Equation 4, we can express \( c \) in terms of \( b \):
\[
c = -\frac{2a^2}{b}
\]
Substituting this into Equation 1 gives:
\[
a + b - \frac{2a^2}{b} = 0
\]
Multiplying through by \( b \) to eliminate the fraction:
\[
ab + b^2 - 2a^2 = 0
\]
This is a quadratic equation in \( b \):
\[
b^2 + ab - 2a^2 = 0
\]
### Step 6: Use the quadratic formula to find \( b \)
Using the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
Here, \( A = 1, B = a, C = -2a^2 \):
\[
b = \frac{-a \pm \sqrt{a^2 + 8a^2}}{2} = \frac{-a \pm 3a}{2}
\]
This gives us two solutions:
1. \( b = a \) (taking the positive root)
2. \( b = -2a \) (taking the negative root)
### Step 7: Find corresponding values of \( c \)
Using \( b = a \) in Equation 1:
\[
a + a + c = 0 \implies c = -2a
\]
Using \( b = -2a \) in Equation 1:
\[
a - 2a + c = 0 \implies c = a
\]
### Step 8: Determine the direction ratios
From the above, we have two sets of direction ratios:
1. \( (a, a, -2a) \) or simplified to \( (1, 1, -2) \)
2. \( (a, -2a, a) \) or simplified to \( (1, -2, 1) \)
### Step 9: Find the angle between the lines
Let \( \mathbf{h_1} = \langle 1, 1, -2 \rangle \) and \( \mathbf{h_2} = \langle 1, -2, 1 \rangle \).
The angle \( \theta \) between the two lines can be found using the formula:
\[
\cos \theta = \frac{\mathbf{h_1} \cdot \mathbf{h_2}}{|\mathbf{h_1}| |\mathbf{h_2}|}
\]
### Step 10: Calculate the dot product and magnitudes
Calculating the dot product:
\[
\mathbf{h_1} \cdot \mathbf{h_2} = 1 \cdot 1 + 1 \cdot (-2) + (-2) \cdot 1 = 1 - 2 - 2 = -3
\]
Calculating the magnitudes:
\[
|\mathbf{h_1}| = \sqrt{1^2 + 1^2 + (-2)^2} = \sqrt{1 + 1 + 4} = \sqrt{6}
\]
\[
|\mathbf{h_2}| = \sqrt{1^2 + (-2)^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6}
\]
### Step 11: Substitute into the cosine formula
\[
\cos \theta = \frac{-3}{\sqrt{6} \cdot \sqrt{6}} = \frac{-3}{6} = -\frac{1}{2}
\]
### Step 12: Find the angle
The angle \( \theta \) is:
\[
\theta = \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \text{ radians}
\]
### Final Answer
The angle between the lines is \( \frac{2\pi}{3} \) radians.
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