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

Step-by-step solution

The equations of the given lines can be rewritten in symmetric form as: L₁: x 1 2 = y 1 3 = z -1 x 3 = y 2 = z -6 L₂: x 1 6 = y -1 = z - 1 4 x 2 = y -12 = z -3 The direction ratios of the lines are proportional to the vectors d ₁ = 3 i + 2 j - 6 k and d ₂ = 2 i - 12 j - 3 k . The angle between the lines is given by = | d ₁ d ₂| | d ₁| | d ₂| . Calculating the dot product: d ₁ d ₂ = (3)(2) + (2)(-12) + (-6)(-3) = 6 - 24 + 18 = 0 Since the dot product is zero, the lines are perpendicular. Thus, the acute angle betwee