2. In the Cartesian coordinate system, $O$ is the origin, $A(1,2)$, $B(3,4)$; in the oblique projection, $A B$ becomes $A^{\prime} B^{\prime}$. Then $\overrightarrow{O A^{\prime}} \cdot \overrightarrow{O B^{\prime}}=$ $\qquad$
$2.5+\frac{5 \sqrt{2}}{2}$.
Note that, in the oblique dimetric projection, the correspondence from the Cartesian coordinate system to the Cartesian coordinate system for coordinates $(x, y)$ is $\left(x+\frac{\sqrt{2} y}{4}, \frac{\sqrt{2} y}{4}\right)$.
Thus, the corresponding points for $(1,2)$ and $(3,4)$ are
$$
\begin{array}{l}
\left(1+\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right),(3+\sqrt{2}, \sqrt{2}) . \\
\text { Hence } \overrightarrow{O A^{\prime}} \cdot \overrightarrow{O B^{\prime}}=\left(1+\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \cdot(3+\sqrt{2}, \sqrt{2}) \\
=5+\frac{5 \sqrt{2}}{2} .
\end{array}
$$