Question

The angle between the lines \( \frac{x}{1} = \frac{y}{1} = \frac{z}{1} \) and \( \frac{x}{0} = \frac{y}{1} = \frac{z}{-1} \) is:

• A. \( \frac{\pi}{2} \)
• B. 0
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{3} \)
• E. \( \sin^{-1}(\sqrt{2}) \)

Hint:

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

Answer 1

The Correct Option is A

The direction ratios of the first line are \( \langle 1, 1, 1 \rangle \), and the direction ratios of the second line are \( \langle 0, 1, -1 \rangle \). 
The angle \( \theta \) between two lines is given by the formula: \[ \cos \theta = \frac{\mathbf{r_1} \cdot \mathbf{r_2}}{|\mathbf{r_1}| |\mathbf{r_2}|} \] Substituting the direction ratios: \[ \cos \theta = \frac{1 \times 0 + 1 \times 1 + 1 \times (-1)}{\sqrt{1^2 + 1^2 + 1^2} \times \sqrt{0^2 + 1^2 + (-1)^2}} = \frac{0 + 1 - 1}{\sqrt{3} \times \sqrt{2}} = 0 \] Thus, \( \theta = \frac{\pi}{2} \).