Question

The angle between the two lines \( \frac{x-4}{1} = \frac{y+4}{2} = \frac{z+1}{2} \) and \( \frac{x+1}{2} = \frac{y+3}{2} = \frac{z-4}{-1} \) is

• A. \( \cos^{-1} \left( \frac{4}{9} \right) \)
• B. \( \cos^{-1} \left( \frac{5}{9} \right) \)
• C. \( \cos^{-1} \left( \frac{1}{9} \right) \)
• D. \( \cos^{-1} \left( \frac{2}{9} \right) \)

Hint:

To find the angle between two lines, use the direction ratios of the lines and apply the formula for the cosine of the angle between them.

Answer 1

The Correct Option is A

Step 1: Use the formula for the angle between two lines.
The direction ratios of the lines are given by: - For the first line, the direction ratios are \( (1, 2, 2) \). - For the second line, the direction ratios are \( (2, 2, -1) \). The formula for the cosine of the angle \( \theta \) between two lines is: \[ \cos \theta = \frac{l_1 \cdot l_2}{|l_1| |l_2|} \] where \( l_1 \) and \( l_2 \) are the direction ratios of the lines.
Step 2: Calculate the cosine of the angle.
Substitute the values of \( l_1 = (1, 2, 2) \) and \( l_2 = (2, 2, -1) \) into the formula: \[ \cos \theta = \frac{1 \cdot 2 + 2 \cdot 2 + 2 \cdot (-1)}{\sqrt{1^2 + 2^2 + 2^2} \cdot \sqrt{2^2 + 2^2 + (-1)^2}} \] This simplifies to: \[ \cos \theta = \frac{4}{9} \]
Step 3: Conclusion.
The angle between the lines is \( \cos^{-1} \left( \frac{4}{9} \right) \).