Question

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

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

Hint:

Convert each line to symmetric form to get its direction ratios, then take the dot product.

Answer 1

The Correct Option is C

Step 1: Scale to integers
Multiplying by 6 and 12 respectively gives direction vectors $(2,3,-6)$ and $(-12,2,-3)$.

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

Step 3: Conclusion
$\cos\theta=0$, so $\theta=\dfrac{\pi}{2}$.

Final Answer:
The angle is 90 degrees. \[ \boxed{\text{(C) }\dfrac{\pi}{2}} \]