If $\overrightarrow{{\mathrm{A}}_1}\mathrm{and}\overrightarrow{{\mathrm{A}}_2}$ are two non-collinear unit vectors and if $|\overrightarrow{A_1}+\overrightarrow{A_2}|=\sqrt{3},$ then the value of $(\overrightarrow{{\mathrm{A}}_1}-\overrightarrow{{\mathrm{A}}_2})·(2\overrightarrow{{\mathrm{A}}_1}+\overrightarrow{{\mathrm{A}}_2})$ is $\frac{3}{2\alpha }.$ The value of $\alpha$ is …….

Here, ${\mathrm{A}}_1={\mathrm{A}}_2=1$

$\mathrm{and}{\mathrm{A}}_1^2+{\mathrm{A}}_2^2+2{\mathrm{A}}_1{\mathrm{A}}_2\mathrm{cos\theta }={\left(\sqrt{3}\right)}^2=3$

$\Rightarrow 1+1+2\times 1\times \mathrm{cos\theta }=3$

$\Rightarrow \mathrm{cos\theta }=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

$\mathrm{Now},(\overrightarrow{{\mathrm{A}}_1}-\overrightarrow{{\mathrm{A}}_2}).(2\overrightarrow{{\mathrm{A}}_1}+\overrightarrow{{\mathrm{A}}_2})=2{\mathrm{A}}_1^2-{\mathrm{A}}_2^2-{\mathrm{A}}_1{\mathrm{A}}_2\mathrm{cos\theta }$

$=2\times 1^2-1^2-1\times 1\times \frac{1}{2}=\frac{1}{2}$

$$\begin{gathered} \frac{1}{2} = \frac{3}{2\mathrm{\alpha }} \\ \Rightarrow \mathrm{\alpha } = 3 \end{gathered}$$