Question

The angle between the lines \( \frac{x+4}{3} = \frac{y-1}{5} = \frac{z+3}{4} \) and \( \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \)

• A. 30°
• B. \( \cos^{-1} \left( \frac{3}{2\sqrt{2}} \right) \)
• C. \( \cos^{-1} \left( \frac{8}{5\sqrt{3}} \right) \)
• D. None of these

Hint:

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

Answer 1

The Correct Option is C

The direction ratios of the first line are \( (3, 5, 4) \), and for the second line, the direction ratios are \( (1, 1, 2) \). The cosine of the angle \( \theta \) between the two lines is given by the formula: \[ \cos \theta = \frac{\mathbf{l_1} \cdot \mathbf{l_2}}{|\mathbf{l_1}| |\mathbf{l_2}|} \] The dot product \( \mathbf{l_1} \cdot \mathbf{l_2} \) is: \[ \mathbf{l_1} \cdot \mathbf{l_2} = 3 \times 1 + 5 \times 1 + 4 \times 2 = 3 + 5 + 8 = 16 \] The magnitudes of the direction ratios are: \[ |\mathbf{l_1}| = \sqrt{3^2 + 5^2 + 4^2} = \sqrt{9 + 25 + 16} = \sqrt{50} \] \[ |\mathbf{l_2}| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \] Now, calculate \( \cos \theta \): \[ \cos \theta = \frac{16}{\sqrt{50} \times \sqrt{6}} = \frac{16}{\sqrt{300}} = \frac{8}{5\sqrt{3}} \] Thus, the angle is \( \cos^{-1} \left( \frac{8}{5\sqrt{3}} \right) \).