(i) To show that the lines do not intersect, equate the components of the general points on l and m:

\(For l: r = 3i − j − 2k + λ(−i + j + 4k)\)

\(For m: r = 4i + 4j − 3k + μ(2i + j − 2k)\)

Equate at least two pairs of components and solve for λ or μ:

\(1. 3 − λ = 4 + 2μ\)

\(2. −1 + λ = 4 + μ\)

Solving these gives λ = \(\frac{3}{2}\) or μ = −\(\frac{7}{2}\).

Verify that not all three pairs of equations are satisfied, confirming the lines do not intersect.

(ii) To calculate the acute angle between the directions of the lines, find the direction vectors:

\(For l: d₁ = −i + j + 4k\)

\(For m: d₂ = 2i + j − 2k\)

\(Calculate the scalar product: d₁ ⋅ d₂ = (−1)(2) + (1)(1) + (4)(−2) = −2 + 1 − 8 = −9\)

Calculate the moduli: |d₁| = \(\sqrt{(-1)^2 + 1^2 + 4^2} = \sqrt{18}\)

|d₂| = \(\sqrt{2^2 + 1^2 + (-2)^2} = \sqrt{9} = 3\)

Divide the scalar product by the product of the moduli and evaluate the inverse cosine:

\(\cos \theta = \frac{-9}{\sqrt{18} \times 3} = \frac{-9}{9\sqrt{2}} = -\frac{1}{\sqrt{2}}\)

\(\theta = \cos^{-1}(-\frac{1}{\sqrt{2}}) = 45^{\circ}\) or \(\frac{1}{4} \pi\) (0.785 radians).