Key Concepts and Formulas

1. Midpoint Property: If a point $P(x_1, y_1, z_1)$ has its mirror image $Q(x_2, y_2, z_2)$ with respect to a plane, then the midpoint $M$ of the line segment $PQ$ lies on the plane. The coordinates of the midpoint are $M\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$.
2. Perpendicularity Property: The line segment $PQ$ joining the point and its mirror image is perpendicular to the plane of reflection. This means the direction vector of the line segment $PQ$ is parallel to the normal vector of the plane. If $\vec{n} = (A, B, C)$ is the normal vector to the plane and $\vec{v} = (x_2-x_1, y_2-y_1, z_2-z_1)$ is the direction vector of $PQ$, then $\frac{x_2-x_1}{A} = \frac{y_2-y_1}{B} = \frac{z_2-z_1}{C} = k$ for some scalar $k$.
3. Plane Equation Conversion: A plane given in vector form $\overrightarrow r \cdot \overrightarrow n = d$ can be converted to Cartesian form $Ax + By + Cz = d$, where $\overrightarrow n = A\widehat i + B\widehat j + C\widehat k$.

Step-by-Step Solution

Let the given point be $P(1, 3, a)$ and its mirror image be $Q(-3, 5, 2)$. The equation of the plane is given as $\overrightarrow r .\left( {2\widehat i - \widehat j + \widehat k} \right) - b = 0$.

Step 1: Convert the Plane Equation to Cartesian Form The given vector equation can be rewritten as $\overrightarrow r .\left( {2\widehat i - \widehat j + \widehat k} \right) = b$. Substituting $\overrightarrow r = x\widehat i + y\widehat j + z\widehat k$, we get the Cartesian form of the plane equation: $(x\widehat i + y\widehat j + z\widehat k) \cdot (2\widehat i - \widehat j + \widehat k) = b$ $2x - y + z = b \quad \text{...(1)}$ The normal vector to this plane is $\overrightarrow n = 2\widehat i - \widehat j + \widehat k$, and its direction ratios are $(2, -1, 1)$.

Step 2: Apply the Midpoint Property The midpoint $M$ of the line segment $PQ$ must lie on the plane $2x - y + z = b$. The coordinates of $M(x_M, y_M, z_M)$ are calculated using the midpoint formula: $M = \left( \frac{x_P + x_Q}{2}, \frac{y_P + y_Q}{2}, \frac{z_P + z_Q}{2} \right)$ Substituting the coordinates of $P(1, 3, a)$ and $Q(-3, 5, 2)$: $M = \left( \frac{1 + (-3)}{2}, \frac{3 + 5}{2}, \frac{a + 2}{2} \right)$ $M = \left( \frac{-2}{2}, \frac{8}{2}, \frac{a + 2}{2} \right)$ $M = \left( -1, 4, \frac{a + 2}{2} \right)$ Since $M$ lies on the plane $2x - y + z = b$, we substitute its coordinates into the plane equation (1): $2(-1) - (4) + \left(\frac{a + 2}{2}\right) = b$ $-2 - 4 + \frac{a + 2}{2} = b$ $-6 + \frac{a + 2}{2} = b$ To eliminate the fraction, multiply the entire equation by 2: $-12 + a + 2 = 2b$ $a - 10 = 2b \quad \text{...(2)}$ This equation provides a relationship between $a$ and $b$.

Step 3: Apply the Perpendicularity Property The line segment $PQ$ is perpendicular to the plane $2x - y + z = b$. This means that the direction vector of $PQ$ must be parallel to the normal vector of the plane. First, find the direction ratios of the line segment $PQ$. We can use $(x_Q - x_P, y_Q - y_P, z_Q - z_P)$: Direction ratios of $PQ = (-3 - 1, 5 - 3, 2 - a)$ $= (-4, 2, 2 - a)$ The direction ratios of the normal vector to the plane $2x - y + z = b$ are $(2, -1, 1)$. Since the line $PQ$ is parallel to the normal vector of the plane, their direction ratios must be proportional: $\frac{-4}{2} = \frac{2}{-1} = \frac{2 - a}{1}$ From the first two ratios, we see that $-2 = -2$, which is consistent. Now, equate the common ratio to the third term: $\frac{2 - a}{1} = -2$ $2 - a = -2$ $a = 4$

Step 4: Solve for 'b' and Calculate $|a + b|$ Now that we have the value of $a = 4$, we can substitute it back into equation (2) to find $b$: $a - 10 = 2b$ $4 - 10 = 2b$ $-6 = 2b$ $b = -3$ Finally, we need to find the value of $|a + b|$: $|a + b| = |4 + (-3)|$ $|a + b| = |4 - 3|$ $|a + b| = |1|$ $|a + b| = 1$

Common Mistakes & Tips

• Sign Errors: Be meticulous with signs, especially when calculating midpoints, direction ratios, or substituting coordinates into the plane equation.
• Plane Equation Form: Ensure the plane equation is correctly converted to Cartesian form ($Ax+By+Cz=d$) and that the normal vector components $(A, B, C)$ and the constant term $d$ are correctly identified.
• Proportionality Consistency: When equating direction ratios, maintain consistency (e.g., numerator from line, denominator from normal, or vice-versa, but do not mix the order).

Summary To find the unknown parameters $a$ and $b$ in this problem, we systematically applied two fundamental properties of mirror images with respect to a plane: the midpoint of the point-image segment lies on the plane, and the point-image segment is perpendicular to the plane. By using these properties, we derived two equations that allowed us to solve for $a=4$ and $b=-3$. Finally, calculating $|a+b|$ yielded $|4+(-3)| = 1$.

The final answer is $\boxed{1}$.