A vector $\overrightarrow{\mathrm{a}}=\mathrm{\alpha }\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\mathrm{\beta }\hat{\mathrm{k}}, (\mathrm{\alpha },\mathrm{\beta }\in \mathrm{R} )$ lies in the plane of the vectors $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+4\hat{\mathrm{k}}.$ If $\overrightarrow{\mathrm{a}}$ bisects the angle between $\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}.$ Then