What is the angle between the lines 2x=3y=-z and 6x=-y=-4z?

  1. A. 0^{\circ}
  2. B. 30^{\circ}
  3. C. 60^{\circ}
  4. D. 90^{\circ} ✓

Correct Answer: D. 90^{\circ}

Explanation

Converting the lines to symmetric form: Line 1 is \frac{x}{3} = \frac{y}{2} = \frac{z}{-6} (dividing by LCM 6), so its DRs are (3, 2, -6). Line 2 is \frac{x}{2} = \frac{y}{-12} = \frac{z}{-3} (dividing by LCM 12), so its DRs are (2, -12, -3). Checking the dot product of DRs: 3(2) + 2(-12) + (-6)(-3) = 6 - 24 + 18 = 0. Since the dot product is 0, the angle between the lines is 90^{\circ}.

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