JEE Main 2024 — 5 April, Shift 2 — Question 7

$|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{c}}|^{2}+|\overrightarrow{\mathrm{a}}|^{2}-2 \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}$

$=|\vec{c}|^{2}+4-0$

$\because \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}$ $|\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|$ $2=3|\overrightarrow{\mathrm{c}}| \sin \alpha$ $|\overrightarrow{\mathrm{c}}|=\frac{2}{3} \operatorname{cosec} \alpha$

$\alpha \in\left[0, \frac{\pi}{3}\right]$ $|\overrightarrow{\mathrm{c}}|_{\min }=\frac{2}{3} \times \frac{2}{\sqrt{3}}$

$\operatorname{cosec} \alpha \in\left[\frac{2}{\sqrt{3}}, \infty\right)$

$\Rightarrow 27|\overrightarrow{\mathrm{c}}-\vec{a}|_{\min }^{2}=27\left(\frac{16}{27}+4\right)=124$