(i) To prove that the lines l and m do not intersect, express the general point of l in component form: \((2 + \lambda, -\lambda, 1 + 2\lambda)\) and the general point of m as \((\mu, 2 + 2\mu, 6 - 2\mu)\). Equate the components and solve for \(\lambda\) and \(\mu\). Possible solutions for \(\lambda\) are -2, \(\frac{1}{4}\), 7, and for \(\mu\) are 0, \(\frac{1}{4}\), -\(\frac{1}{2}\). Verify that all three component equations are not satisfied simultaneously, confirming that the lines do not intersect.

Alternatively, use the scalar triple product: \((2i - 2j - 5k) \cdot ((i - j + 2k) \times (i + 2j - 2k))\). Calculate the determinant to obtain a non-zero value, e.g., -27, indicating the lines do not intersect.

(ii) To find the acute angle between the directions of l and m, calculate the scalar product of the direction vectors \((i - j + 2k)\) and \((i + 2j - 2k)\). The scalar product is -3. Calculate the moduli of the direction vectors: \(\sqrt{6}\) and \(\sqrt{9}\). Use the formula \(\cos \theta = \frac{-3}{\sqrt{6} \times \sqrt{9}}\) to find \(\theta\). The acute angle is approximately 47.1° or 0.822 radians.