To find the value of a, we express the general points of lines l and m in component form:

\(Line l: r = (a + λ)i + (2 - 2λ)j + (3 + 3λ)k\)

\(Line m: r = (2 + 2μ)i + (1 - μ)j + (2 + μ)k\)

Since the lines intersect, equate the corresponding components:

\(1. a + λ = 2 + 2μ\)

\(2. 2 - 2λ = 1 - μ\)

\(3. 3 + 3λ = 2 + μ\)

From equation (2):

\(μ = 2 - 2λ - 1 = 1 - 2λ\)

\(Substitute μ = 1 - 2λ into equation (1):\)

\(a + λ = 2 + 2(1 - 2λ)\)

\(a + λ = 2 + 2 - 4λ\)

\(a + λ = 4 - 4λ\)

\(5λ = 4 - a\)

\(λ = \frac{4 - a}{5}\)

\(Substitute λ = \frac{4 - a}{5} into equation (3):\)

\(3 + 3\left(\frac{4 - a}{5}\right) = 2 + (1 - 2\left(\frac{4 - a}{5}\right))\)

\(3 + \frac{12 - 3a}{5} = 2 + 1 - \frac{8 - 2a}{5}\)

\(3 + \frac{12 - 3a}{5} = 3 - \frac{8 - 2a}{5}\)

Multiply through by 5 to clear fractions:

\(15 + 12 - 3a = 15 - 8 + 2a\)

\(27 - 3a = 7 + 2a\)

\(5a = 20\)

\(a = -6\)