Consider the line `L_(1) : (x-1)/(2)=(y)/(-1)=(z+3)/(1), L_(2) : (x-4)/(1)=(y+3)/(1)=(z+3)/(2)` find the angle between them.

Answer

To find the angle between the two lines \( L_1 \) and \( L_2 \), we can use the formula for the cosine of the angle \( \theta \) between two lines in three-dimensional space. The lines are given in symmetric form, and we can extract the direction ratios from them.

### Step 1: Identify the direction ratios of the lines

The line \( L_1 \) is given by:
\[ \frac{x - 1}{2} = \frac{y}{-1} = \frac{z + 3}{1} \]
From this, we can identify the direction ratios \( a_1, b_1, c_1 \) as:
- \( a_1 = 2 \)
- \( b_1 = -1 \)
- \( c_1 = 1 \)

The line \( L_2 \) is given by:
\[ \frac{x - 4}{1} = \frac{y + 3}{1} = \frac{z + 3}{2} \]
From this, we can identify the direction ratios \( a_2, b_2, c_2 \) as:
- \( a_2 = 1 \)
- \( b_2 = 1 \)
- \( c_2 = 2 \)

### Step 2: Use the formula for the cosine of the angle between two lines

The formula for the cosine of the angle \( \theta \) between the two lines is given by:
\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

### Step 3: Calculate the numerator

Substituting the values of \( a_1, b_1, c_1, a_2, b_2, c_2 \):
\[ \text{Numerator} = a_1 a_2 + b_1 b_2 + c_1 c_2 = (2)(1) + (-1)(1) + (1)(2) \]
Calculating this gives:
\[ = 2 - 1 + 2 = 3 \]

### Step 4: Calculate the denominator

Now we calculate \( \sqrt{a_1^2 + b_1^2 + c_1^2} \) and \( \sqrt{a_2^2 + b_2^2 + c_2^2} \):
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{2^2 + (-1)^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \]
Thus, the denominator becomes:
\[ \sqrt{6} \cdot \sqrt{6} = 6 \]

### Step 5: Calculate \( \cos \theta \)

Now substituting back into the formula:
\[ \cos \theta = \frac{3}{6} = \frac{1}{2} \]

### Step 6: Find the angle \( \theta \)

To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) \]
This gives:
\[ \theta = \frac{\pi}{3} \text{ radians} \quad \text{or} \quad 60^\circ \]

### Final Answer
The angle between the lines \( L_1 \) and \( L_2 \) is \( \frac{\pi}{3} \) radians or \( 60^\circ \).

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