$\begin{aligned} & |\vec{c}-\vec{a}|=|\vec{c}|^2+|\vec{a}|^2-2 \bar{a} . \bar{c} \\ & =|\vec{c}|^2+4-0 \\ & \because \vec{a}=\vec{b} \times \vec{c} \\ & |\vec{a}|=|\vec{b} \times \vec{c}| \\ & 2=3|\vec{c}| \sin \alpha \\ & |\vec{c}|=\frac{2}{3} \operatorname{cosec} \alpha \\ & \alpha \in\left[0, \frac{\pi}{3}\right] \\ & |\vec{c}|_{\min }=\frac{2}{3} \times \frac{2}{\sqrt{3}}\end{aligned}$
$\begin{aligned} & \operatorname{cosec} \alpha \in\left[\frac{2}{\sqrt{3}}, \infty\right) \\ & \Rightarrow 27|\vec{c}-\vec{a}|_{\min }^2=27\left(\frac{16}{27}+4\right)=124\end{aligned}$