$\overline{\mathrm{\alpha }}={\mathrm{a}}_1\overline{\mathrm{i}}+{\mathrm{a}}_2\overline{\mathrm{j}}+{\mathrm{a}}_3\overline{\mathrm{k}}$ where ${\mathrm{a}}_1, {\mathrm{a}}_2, {\mathrm{a}}_3$are ratio- nal numbers. If $\overline{\mathrm{\alpha }}$ makes an angle ${40}^0$ with the vector $\overline{\mathrm{\beta }}=\sqrt{2}\mathrm{i}+3\sqrt{2}\mathrm{j}+4\overline{\mathrm{k}}$ and $\left|\overline{\mathrm{\alpha }}\right|$ is rational, then $\left|\overline{\mathrm{\alpha }}\right|$ lies in