根据题意有\[\begin{split}\sum_{k=0}^{11}\left(\vec a_k\cdot \vec a_{k+1}\right)&=\sum_{k=0}^{11}\left[\cos\dfrac k6\pi\cos\dfrac{k+1}{6}\pi+\left(\sin\dfrac{k}{6}\pi+\cos\dfrac{k}{6}\pi\right)\left(\sin\dfrac{k+1}{6}\pi+\cos\dfrac{k+1}{6}\pi\right)\right]\\&=\sum_{k=0}^{11}\left[\cos\dfrac k6\pi\left(\dfrac{\sqrt 3}2\cos\dfrac k6\pi-\dfrac 12\sin\dfrac k6\pi\right)+\right.\\&\left.\qquad\qquad\left(\sin\dfrac k6\pi+\cos\dfrac k6\pi\right)\left(\dfrac{\sqrt 3-1}2\sin\dfrac k6\pi+\dfrac{\sqrt 3+1}2\cos\dfrac k6\pi\right)\right],\end{split}\]注意到\[\begin{split}&\sum_{k=0}^{11}\cos^2\dfrac k6\pi=\sum_{k=0}^{11}\dfrac{1+\cos\dfrac k3\pi}{2}=6,\\&\sum_{k=0}^{11}\cos\dfrac k6\pi\sin\dfrac k6\pi=\sum_{k=0}^{11}\dfrac 12\sin\dfrac k3\pi=0,\\&\sum_{k=0}^{11}\sin^2\dfrac k6\pi=\sum_{k=0}^{11}\dfrac{1-\cos \dfrac k3\pi}{2}=6,\end{split}\]于是不难计算得所求和式的值为 $9\sqrt 3$．