The acute angle between the lines (x=-2+2 t , y =3-4 t , z =-4+ t ) and (x=-2- t , y =3+2 t , z =-4+3 t ) is

Step-by-step solution

Determine the direction vectors from parametric equations The first line with parametric form x = -2 + 2t , y = 3 - 4t , z = -4 + t yields direction vector d₁ = 2, -4, 1 . The second line x = -2 - t , y = 3 + 2t , z = -4 + 3t gives d₂ = -1, 2, 3 . Apply the angle formula between direction vectors The acute angle satisfies = | d₁ d₂ | | d₁ | | d₂ | . Compute the dot product and magnitudes: d₁ d₂ = (2)(-1) + (-4)(2) + (1)(3) = -7 | d₁ | = 2^2 + (-4)^2 + 1^2 = 21 | d₂ | = (-1)^2 + 2^2 + 3^2 = 14 Substitute into the fo