5. Given vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are two perpendicular unit vectors in a plane, and $(3 a-c) \cdot(4 b-c)=0$. Then the maximum value of $|c|$ is $\qquad$ .
5.5.

Let $\overrightarrow{O A}=3 a, \overrightarrow{O B}=4 b, \overrightarrow{O C}=c$.
From the given, $\overrightarrow{A C} \perp \overrightarrow{B C}$.
Thus, point $C$ lies on the circle with $A B=5$ as its diameter, and this circle passes through the origin $O$.
Therefore, the maximum value of $|\overrightarrow{O C}|=|c|$ is 5.