Step 1: Understanding the Question:
We need to find the coordinates of the mirror image of a point with respect to a line and then find the sum of these coordinates.
Step 3: Detailed Explanation:
1. Find the equation of line L through A(1, 3, 2) and B(2, 2, 1):
Direction ratios are $(2-1, 2-3, 1-2) = (1, -1, -1)$.
Equation of line: $\vec{r} = (1, 3, 2) + \lambda(1, -1, -1)$.
2. Let M be the foot of the perpendicular from P(1, 1, -1) to the line. M is $(1+\lambda, 3-\lambda, 2-\lambda)$.
3. Vector $\vec{PM} = M - P = (\lambda, 2-\lambda, 3-\lambda)$.
4. $\vec{PM}$ is perpendicular to line direction $(1, -1, -1)$:
$1(\lambda) - 1(2-\lambda) - 1(3-\lambda) = 0 \implies \lambda - 2 + \lambda - 3 + \lambda = 0 \implies 3\lambda = 5 \implies \lambda = 5/3$.
5. Coordinates of M: $(1 + 5/3, 3 - 5/3, 2 - 5/3) = (8/3, 4/3, 1/3)$.
6. M is the midpoint of P(1, 1, -1) and image (x, y, z):
$\frac{x+1}{2} = 8/3 \implies x = 16/3 - 1 = 13/3$
$\frac{y+1}{2} = 4/3 \implies y = 8/3 - 1 = 5/3$
$\frac{z-1}{2} = 1/3 \implies z = 2/3 + 1 = 5/3$
7. Sum $x + y + z = \frac{13+5+5}{3} = \frac{23}{3}$.
Step 4: Final Answer:
The sum $x + y + z$ is $\frac{23}{3}$.