If $$(a, b, c)$$ is the image of the point $$(1, 2, -3)$$ in the line, $$\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z}{-1}$$, then $$a + b + c$$ is equal to:

Let us denote the given point as $$P(1, 2, -3)$$, and let $$F$$ be the foot of the perpendicular dropped from $$P$$ onto the given straight line.

The equation of the given line is:

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

Any general point on this line can represent the coordinates of the foot $$F$$ in terms of the scalar parameter $$\lambda$$:

$$F = (2\lambda - 1, -2\lambda + 3, -\lambda)$$

Now, let us find the direction ratios of the line segment $$PF$$ by subtracting the coordinates of point $$P$$ from point $$F$$:

$$\text{Direction Ratios of } PF = ((2\lambda - 1) - 1, (-2\lambda + 3) - 2, -\lambda - (-3))$$

$$\text{Direction Ratios of } PF = (2\lambda - 2, -2\lambda + 1, -\lambda + 3)$$

The direction ratios of the given line are the denominators of the symmetric equation, which are $$(2, -2, -1)$$. Since the line segment $$PF$$ is perpendicular to the given line, the dot product of their direction ratios must equal zero:

$$2(2\lambda - 2) + (-2)(-2\lambda + 1) + (-1)(-\lambda + 3) = 0$$

Expanding the terms and simplifying the equation:

$$(4\lambda - 4) + (4\lambda - 2) + (\lambda - 3) = 0$$

$$4\lambda + 4\lambda + \lambda - 4 - 2 - 3 = 0$$

$$9\lambda - 9 = 0 \implies 9\lambda = 9 \implies \lambda = 1$$

Substitute the value of $$\lambda = 1$$ back into our expression for the foot $$F$$ to find its exact coordinates:

$$F = (2(1) - 1, -2(1) + 3, -(1)) = (1, 1, -1)$$

The image point is given as $$I(a, b, c)$$. Geometrically, the foot of the perpendicular $$F$$ acts exactly as the midpoint of the line segment connecting the original point $$P$$ to its image $$I$$.

Using the midpoint formula:

$$F = \frac{P + I}{2} \implies I = 2F - P$$

Let us calculate each coordinate of the image point $$(a, b, c)$$ separately:

$$a = 2(1) - 1 = 1$$

$$b = 2(1) - 2 = 0$$

$$c = 2(-1) - (-3) = -2 + 3 = 1$$

Thus, the coordinates of the image point are $$(a, b, c) = (1, 0, 1)$$.

The question asks for the value of $$a + b + c$$:

$$a + b + c = 1 + 0 + 1 = 2$$