Let the mirror image of the point $$(1, 3, a)$$ with respect to the plane $$\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) - b = 0$$ be $$(-3, 5, 2)$$. Then the value of $$|a + b|$$ is equal to ___.

Let the point be $$A = (1, 3, a)$$ and its mirror image be $$B = (-3, 5, 2)$$ with respect to the plane $$2x - y + z = b$$. The line $$AB$$ must be perpendicular to the plane, so the direction $$\vec{AB} = (-4, 2, 2-a)$$ is parallel to the normal $$\vec{n} = (2, -1, 1)$$.

Setting up proportionality: $$\dfrac{-4}{2} = \dfrac{2}{-1} = \dfrac{2-a}{1}$$, which gives $$-2 = -2 = 2 - a$$, so $$a = 4$$.

The midpoint of $$A$$ and $$B$$ must lie on the plane. The midpoint is $$\left(\dfrac{1+(-3)}{2},\,\dfrac{3+5}{2},\,\dfrac{4+2}{2}\right) = (-1, 4, 3)$$. Substituting into the plane equation: $$2(-1) - 4 + 3 = -3 = b$$.

Therefore $$|a + b| = |4 + (-3)| = |1| = 1$$.