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

Step-by-step solution

Angle Between Two Lines The direction vectors are determined from the symmetric forms of the lines. For the first line 3x = 2y = -z , the direction vector simplifies to d₁ = (2, 3, -6) after clearing denominators. For the second line -x = 6y = -4z , the direction vector becomes d₂ = (-12, 2, -3) . The angle between the lines is given by = | d₁ d₂ | | d₁ | | d₂ | . Computing the dot product: d₁ d₂ = (2)(-12) + (3)(2) + (-6)(-3) = -24 + 6 + 18 = 0 . Since the dot product is zero, the lines are perpendicular and = 2 .