Let  $\overline{a}$  and  $\overline{b}$  be two unit vectors such that  $|\overline{a}+\overline{b}|=\sqrt{3}$ . If $\overline{c}=\overline{a}+2\overline{b}+3(\overline{a}\times \overline{b})$  , then $2\left|\overline{c}\right|$is equal to

$$\begin{gathered} |\overline{a}+\overline{b}|=\sqrt{3} angle between \overline{a} \& \overline{b} is {60}^0 \\ \Rightarrow \overline{a}\times \overline{b} is perpendicular to plane containing \overline{a} \& \overline{b} \\ \Rightarrow \overline{c}=\overline{a}+2\overline{b}+3(\overline{a}\times \overline{b}) \overline{c}=|\overline{a}+2\overline{b}+3(\overline{a}\times \overline{b})| \\ =1+4\left(1\right)+9{\left\{\mathrm{1.1.}\frac{\sqrt{3}}{2}\right\}}^2+2(2\left|a\right|\left|b\right|\cos {60}^0+6\left(0\right)+3\left(0\right)) \\ =1+4+9\left(\frac{3}{4}\right)+2(\mathrm{2.1.1.}\frac{1}{2}+0) =1+4+\frac{27}{4}+2\left(1\right) =7+\frac{27}{4}=\frac{55}{4} \\ \left|\overline{c}\right|=\sqrt{\frac{55}{4}}=\frac{\sqrt{55}}{2} \Rightarrow 2\left|\overline{c}\right|=\sqrt{55} \end{gathered}$$