Question

The angle between the lines \( \frac{x - 1}{4} = \frac{y - 3}{1} = \frac{z}{8} \) and \( \frac{x - 2}{2} = \frac{y + 1}{2} = \frac{z - 4}{1} \) is

• A. \( \sin^{-1} \left( \frac{2}{3} \right) \)
• B. \( \cos^{-1} \left( \frac{2}{3} \right) \)
• C. \( \cos^{-1} \left( \frac{1}{3} \right) \)
• D. \( \sin^{-1} \left( \frac{1}{3} \right) \)

Hint:

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

Answer 1

The Correct Option is B

Step 1: Find the direction ratios of the lines.
For the first line, the direction ratios are \( (4, 1, 8) \) and for the second line, the direction ratios are \( (2, 2, 1) \).
Step 2: Use the formula for the angle between two lines.
The formula for the angle between two lines with direction ratios \( \vec{a} = (a_1, a_2, a_3) \) and \( \vec{b} = (b_1, b_2, b_3) \) is: \[ \cos \theta = \frac{a_1 b_1 + a_2 b_2 + a_3 b_3}{\sqrt{a_1^2 + a_2^2 + a_3^2} \sqrt{b_1^2 + b_2^2 + b_3^2}} \] Step 3: Calculation.
Substitute the direction ratios into the formula, we get \( \cos \theta = \frac{2}{3} \), so \( \theta = \cos^{-1} \left( \frac{2}{3} \right) \).
Step 4: Conclusion.
The correct answer is (B) \( \cos^{-1} \left( \frac{2}{3} \right) \).