The line $$l_1$$ passes through the point $$(2, 6, 2)$$ and is perpendicular to the plane $$2x + y - 2z = 10$$. Then the shortest distance between the line $$l_1$$ and the line $$\frac{x+1}{2} = \frac{y+4}{-3} = \frac{z}{2}$$ is:

The line $$l_1$$ passes through the point $$A(2, 6, 2)$$ and is perpendicular to the plane $$2x + y - 2z = 10$$. Therefore, its direction vector $$\vec{d}_1$$ is equal to the normal vector of the plane:

$$\vec{d}_1 = 2\hat{i} + \hat{j} - 2\hat{k}$$

The second line $$l_2$$ is given by:

$$\frac{x+1}{2} = \frac{y+4}{-3} = \frac{z}{2}$$

From this equation, a point passing through $$l_2$$ is $$B(-1, -4, 0)$$ and its direction vector is:

$$\vec{d}_2 = 2\hat{i} - 3\hat{j} + 2\hat{k}$$

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Step 1: Form the vector connecting the two points $$A$$ and $$B$$

The vector $$\vec{AB}$$ is calculated as:

$$\vec{AB} = (-1 - 2)\hat{i} + (-4 - 6)\hat{j} + (0 - 2)\hat{k} = -3\hat{i} - 10\hat{j} - 2\hat{k}$$

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Step 2: Calculate the cross product of the direction vectors $$\vec{d}_1 \times \vec{d}_2$$

$$\vec{d}_1 \times \vec{d}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -2 \\ 2 & -3 & 2 \end{vmatrix}$$

$$= \hat{i}(1(2) - (-2)(-3)) - \hat{j}(2(2) - (-2)(2)) + \hat{k}(2(-3) - 1(2))$$

$$= -4\hat{i} - 8\hat{j} - 8\hat{k}$$

Finding the magnitude of this cross product vector:

$$\left|\vec{d}_1 \times \vec{d}_2\right| = \sqrt{(-4)^2 + (-8)^2 + (-8)^2} = \sqrt{16 + 64 + 64} = \sqrt{144} = 12$$

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Step 3: Evaluate the shortest distance using the scalar triple product

The shortest distance $$d$$ between two skew lines is given by the formula:

$$d = \frac{\left|\vec{AB} \cdot (\vec{d}_1 \times \vec{d}_2)\right|}{\left|\vec{d}_1 \times \vec{d}_2\right|}$$

Computing the dot product in the numerator:

$$\vec{AB} \cdot (\vec{d}_1 \times \vec{d}_2) = (-3)(-4) + (-10)(-8) + (-2)(-8)$$

$$\vec{AB} \cdot (\vec{d}_1 \times \vec{d}_2) = 12 + 80 + 16 = 108$$

Substituting the values into the shortest distance formula:

$$d = \frac{108}{12} = 9$$

Therefore, the shortest distance between the lines is equal to 9.