1. Given that the hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ is a regular hexagon inscribed in a circle $\odot O$ with radius $R$. Establish a mapping $f: P \rightarrow f(P)$ from a point $P$ on the plane $\alpha$ of $\odot O$ to the set of real numbers $\mathbf{R}$, where $f(P)=\left|\sum_{i=1}^{6} P A_{i}\right|$. When the point $P$ varies inside or on the circumference of $\odot O$, the maximum value of $f(P)$ is $(\quad)$.
(A) $3 R$
(B) $6 R$
(C) $9 R$
(D) $12 R$
1.1.B.
Since the hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ is a regular hexagon with center $O$, we have $\sum_{i=1}^{6} O A_{i}=0$.
$$
\begin{array}{l}
\text { Hence } f(P)=\left|\sum_{i=1}^{6} P A_{i}\right|=\left|\sum_{i=1}^{6}\left(P O+\boldsymbol{O A}_{i}\right)\right| \\
=\left|\sum_{i=1}^{6} \boldsymbol{P O}\right|=6|P O| \leqslant 6 R .
\end{array}
$$