Question

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

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

Hint:

Convert each line to symmetric form to read its direction ratios, then test the dot product.

Answer 1

The Correct Option is D

Step 1: Read the directions by comparing with the standard form:
Divide $2x = 3y = -z$ by 6: $\frac{x}{3} = \frac{y}{2} = \frac{z}{-6}$. Direction $(3, 2, -6)$.
Divide $6x = -y = -4z$ by 12: $\frac{x}{2} = \frac{y}{-12} = \frac{z}{-3}$. Direction $(2, -12, -3)$.

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

Step 3: Conclusion:
A zero dot product means the lines are at right angles. The "acute angle" asked for is then $\frac\pi2$ itself.

Final Answer:
Option (D). \[ \boxed{\frac{\pi}{2} \text{ (D)}} \]