11. Given that vectors $\alpha, \beta$ are two mutually perpendicular unit vectors in a plane, and
$$
(3 \boldsymbol{\alpha}-\boldsymbol{\gamma}) \cdot(4 \boldsymbol{\beta}-\boldsymbol{\gamma})=0 .
$$

Then the maximum value of $|\boldsymbol{\gamma}|$ is $\qquad$
11. 5 .

As shown in Figure 2, let
$$
\begin{array}{l}
\overrightarrow{O A}=3 \alpha, \\
\overrightarrow{O B}=4 \boldsymbol{\beta}, \\
\overrightarrow{O C}=\gamma .
\end{array}
$$

From the given information, $\overrightarrow{A C} \perp \overrightarrow{B C}$. Therefore, point $C$ lies on the circle with $A B$ as its diameter, and this circle passes through the origin.
Thus, the maximum value of $|\overrightarrow{O C}|$ is 5.