To solve the problem, we need to find the values of \( a \) and \( b \) based on the given information about the point and the plane. Let's break it down step by step.
### Step 1: Understand the Given Information
We have a point \( P(1, 3, a) \) and its image in the plane is given as \( Q(-3, 5, 2) \). The equation of the plane is given in vector form as \( \vec{r} \cdot (2 \hat{i} - \hat{j} + \hat{k}) - b = 0 \).
### Step 2: Convert the Plane Equation to Cartesian Form
The vector equation can be converted to Cartesian form. The normal vector of the plane is \( \vec{n} = (2, -1, 1) \). The Cartesian equation of the plane can be written as:
\[
2x - y + z = b
\]
### Step 3: Find the PQ Vector
The vector \( \vec{PQ} \) from point \( P \) to point \( Q \) is calculated as:
\[
\vec{PQ} = Q - P = (-3 - 1, 5 - 3, 2 - a) = (-4, 2, 2 - a)
\]
### Step 4: Set Up the Proportionality Condition
Since the vector \( \vec{PQ} \) is parallel to the normal vector \( \vec{n} \), we can set up the following proportionality:
\[
\frac{-4}{2} = \frac{2}{-1} = \frac{2 - a}{1}
\]
### Step 5: Solve for \( a \)
From the first two ratios, we have:
\[
\frac{-4}{2} = -2 \quad \text{and} \quad \frac{2}{-1} = -2
\]
Thus, we can equate:
\[
\frac{2 - a}{1} = -2
\]
Solving for \( a \):
\[
2 - a = -2 \implies a = 4
\]
### Step 6: Find the Midpoint \( R \)
The midpoint \( R \) of points \( P \) and \( Q \) can be calculated as:
\[
R = \left( \frac{1 + (-3)}{2}, \frac{3 + 5}{2}, \frac{a + 2}{2} \right) = \left( \frac{-2}{2}, \frac{8}{2}, \frac{4 + 2}{2} \right) = (-1, 4, 3)
\]
### Step 7: Substitute \( R \) into the Plane Equation
Now, we substitute \( R(-1, 4, 3) \) into the plane equation to find \( b \):
\[
2(-1) - 4 + 3 = b
\]
Calculating this gives:
\[
-2 - 4 + 3 = b \implies b = -3
\]
### Step 8: Calculate \( |a + b| \)
Now we have \( a = 4 \) and \( b = -3 \). We need to find \( |a + b| \):
\[
|a + b| = |4 - 3| = |1| = 1
\]
### Final Answer
Thus, the final answer is:
\[
\boxed{1}
\]
