$$\begin{gathered} \mathrm{If} \overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}} \mathrm{are} \mathrm{three} \mathrm{vectors} \mathrm{such} \mathrm{that} \\ \left|\overline{\mathrm{a}}\right|=\left|\overline{\mathrm{c}}\right|=1, \left|\overline{\mathrm{b}}\right|=4, \left|\overline{\mathrm{b}}\times \overline{\mathrm{c}}\right|=2 \mathrm{and} 2\overline{\mathrm{b}}=\overline{\mathrm{c}}+\mathrm{\lambda }\overline{\mathrm{a}} \\ \mathrm{then} \mathrm{\lambda }= \end{gathered}$$

$$\begin{gathered} |\overline{\mathrm{b}}\times \overline{\mathrm{c}}|=2 \Rightarrow |\overline{\mathrm{b}}\times \overline{\mathrm{c}}{|}^2+{(\overline{\mathrm{b}}·\overline{\mathrm{c}})}^2=|\overline{\mathrm{b}}{|}^2 |\overline{\mathrm{c}}{|}^2 \\ \Rightarrow 4+{(\overline{\mathrm{b}}·\overline{\mathrm{c}})}^2=16\times 1 \\ \Rightarrow {(\overline{\mathrm{b}}·\overline{\mathrm{c}})}^2=12 2\overline{\mathrm{b}}-\overline{\mathrm{c}}=\mathrm{\lambda }\overline{\mathrm{a}} \Rightarrow |2\overline{\mathrm{b}}-\overline{\mathrm{c}}{|}^2={\mathrm{\lambda }}^2|\overline{\mathrm{a}}{|}^2 \\ \Rightarrow 4\times 16+1-4\sqrt{12}={\mathrm{\lambda }}^2\left(1\right) \\ \Rightarrow \mathrm{\lambda }=\sqrt{65-8\sqrt{3}} \end{gathered}$$