$$\begin{gathered} \mathrm{Let} \stackrel{\rightharpoonup }{\mathrm{a}} \mathrm{and} \stackrel{\rightharpoonup }{\mathrm{b}} \mathrm{be} \mathrm{two} \mathrm{non}-\mathrm{zero} \mathrm{vectors} \\ \mathrm{perpendicular} \mathrm{to} \mathrm{each} \mathrm{other} \mathrm{and} \left|\stackrel{\rightharpoonup }{\mathrm{a}}\right|=\left|\stackrel{\rightharpoonup }{\mathrm{b}}\right|. \\ \mathrm{If} \left|\stackrel{\rightharpoonup }{\mathrm{a}}\times \stackrel{\rightharpoonup }{\mathrm{b}}\right|=\left|\stackrel{\rightharpoonup }{\mathrm{a}}\right|, \mathrm{then} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{the} \\ \mathrm{vectors} \left(\stackrel{\rightharpoonup }{\mathrm{a}}+\stackrel{\rightharpoonup }{\mathrm{b}}+\left(\stackrel{\rightharpoonup }{\mathrm{a}}\times \stackrel{\rightharpoonup }{\mathrm{b}}\right)\right) \mathrm{and} \stackrel{\rightharpoonup }{\mathrm{a}} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

$\begin{array}{rcl}\left|\overline{\mathrm{a}}\right|& =& \left|\overline{\mathrm{b}}\right|, \overline{\mathrm{a}}.\overline{\mathrm{b}}=0\\ \left|\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right|& =& \left|\overline{\mathrm{a}}\right|\Rightarrow \left|\overline{\mathrm{a}}\right|\left|\overline{\mathrm{b}}\right| \sin {90}^{°}=\left|\overline{\mathrm{b}}\right|\Rightarrow \left|\overline{\mathrm{a}}\right|=1\\ {\left|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right|}^2& =& {\left|\overline{\mathrm{a}}\right|}^2+{\left|\overline{\mathrm{b}}\right|}^2+{\left|\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right|}^2+0+0+0\\ & =& 1+1+1=3\\ \left|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right|& =& \sqrt{3}\\ \cos \mathrm{\theta }& =& \frac{(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{a}}\times \overline{\mathrm{b}}).\overline{\mathrm{a}}}{\left|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right|\left|\overline{\mathrm{a}}\right|}=\frac{1+0+0}{\sqrt{3}\times 1}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.\end{array}$