Question

The angle between the lines \( 2x = 3y = -z \) and \( 6x = -y = -4z \) is

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{2} \)
• E. \( \frac{2\pi}{3} \)

Hint:

If dot product = 0, lines are perpendicular ⇒ angle = \( \frac{\pi}{2} \).

Answer 1

The Correct Option is A

Concept:
• Convert symmetric form to direction ratios.
• Use: \[ \cos\theta = \frac{\vec{d_1}\cdot\vec{d_2}}{|\vec{d_1}||\vec{d_2}|} \]

Step 1: Convert first line.
\[ 2x = 3y = -z = t \] \[ x = \frac{t}{2}, \; y = \frac{t}{3}, \; z = -t \] Direction ratios: \[ \left(\frac{1}{2}, \frac{1}{3}, -1\right) \] Multiply by 6: \[ (3,2,-6) \]

Step 2: Second line.
\[ 6x = -y = -4z = s \] \[ x = \frac{s}{6}, y = -s, z = -\frac{s}{4} \] Direction ratios: \[ \left(\frac{1}{6}, -1, -\frac{1}{4}\right) \] Multiply by 12: \[ (2,-12,-3) \]

Step 3: Dot product.
\[ 3\cdot2 + 2(-12) + (-6)(-3) \] \[ = 6 - 24 + 18 = 0 \]

Step 4: Conclusion.
\[ \cos\theta = 0 \Rightarrow \theta = \frac{\pi}{2} \]

Step 5: Final Answer.
\[ \boxed{\frac{\pi}{2}} \]