$ =|\overrightarrow{\mathrm{c}}|^2+4-0 $
$ \because \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} $
$ |\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}| $
$ 2=3|\overrightarrow{\mathrm{c}}| \sin \alpha $
$ |\overrightarrow{\mathrm{c}}|=\frac{2}{3} \operatorname{cosec} \alpha \quad \alpha \in\left[0, \frac{\pi}{3}\right] $
$ |\overrightarrow{\mathrm{c}}|_{\min }=\frac{2}{3} \times \frac{2}{\sqrt{3}} \quad \operatorname{cosec} \alpha \in\left[\frac{2}{\sqrt{3}}, \infty\right) $
$ \Rightarrow 27|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|_{\min }^2=27\left(\frac{16}{27}+4\right)=124$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.