If $\overline{\mathrm{a}} . \overline{\mathrm{b}} =0$ and $\overline{\mathrm{a}} + \overline{\mathrm{b}}$ makes an angle of ${30}^{\mathrm{o}}$ with $\overline{\mathrm{a}},$ then

$$\begin{gathered} {\left|\overline{\mathrm{a}}+\overline{\mathrm{b}}\right|}^2={\left|\overline{\mathrm{a}}\right|}^2+{\left|\overline{\mathrm{b}}\right|}^2+2\overline{\mathrm{a}}.\overline{\mathrm{b}}={\left|\overline{\mathrm{a}}\right|}^2+{\left|\overline{\mathrm{b}}\right|}^2 \\ \frac{\sqrt{3}}{2}=\frac{(\overline{\mathrm{a}}+\overline{\mathrm{b}})}{\left|\overline{\mathrm{a}}+\overline{\mathrm{b}}\right| \left|\overline{\mathrm{a}}\right|} \Rightarrow \sqrt{3} \left|\overline{\mathrm{a}}+\overline{\mathrm{b}}\right| \left|\overline{\mathrm{a}}\right|=2{\overline{\left|\overline{\mathrm{a}}\right|}}^2 \\ \mathrm{S}.\mathrm{B}.\mathrm{S} \Rightarrow 3\left({\left|\overline{\mathrm{a}}\right|}^2+{\left|\overline{\mathrm{b}}\right|}^2\right)=4{\left|\overline{\mathrm{a}}\right|}^2 \\ \Rightarrow 3{\left|\overline{\mathrm{b}}\right|}^2={\left|\overline{\mathrm{a}}\right|}^2 \\ \Rightarrow \sqrt{3}\left|\overline{\mathrm{b}}\right| =\left|\overline{\mathrm{a}}\right| \end{gathered}$$