Question

The angle between the lines \[ \frac{x - 1}{4} = \frac{y - 3}{8} = \frac{z - 2}{2} \quad \text{and} \quad \frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 4}{2} \] is

• A. \( \cos^{-1} \left( \frac{3}{4} \right) \)
• B. \( \cos^{-1} \left( \frac{1}{3} \right) \)
• C. \( \cos^{-1} \left( \frac{1}{2} \right) \)
• D. \( \cos^{-1} \left( \frac{2}{3} \right) \)

Hint:

To find the angle between two lines, use the formula involving their direction ratios and apply the dot product.

Answer 1

The Correct Option is D

Step 1: Find the direction cosines of the lines.
We are given the direction ratios of two lines. For the first line, the direction ratios are \( \vec{l_1} = (4, 8, 2) \), and for the second line, the direction ratios are \( \vec{l_2} = (3, 4, 2) \).
Step 2: Use the formula for the angle between two lines.
The formula for the angle \( \theta \) between two lines with direction cosines \( \vec{l_1} \) and \( \vec{l_2} \) is: \[ \cos \theta = \frac{\vec{l_1} \cdot \vec{l_2}}{|\vec{l_1}| |\vec{l_2}|}. \] Substitute the values and simplify to find \( \theta = \cos^{-1} \left( \frac{2}{3} \right) \).
Step 3: Conclusion.
Thus, the correct answer is \( \cos^{-1} \left( \frac{2}{3} \right) \), corresponding to option (D).