$|\hat d|\,\, = \,\,\sqrt {a_1^2\,\, + \,\,a_2^2 + \,\,a_3^3} \,\, = \,\,1$
$ \Rightarrow a_1^2\,\, + \,\,a_2^2\, + \,a_3^2\, = \,\,1\,\,\,\,\,.......\,\,(i)\,\,;\,$
$a\,.\,\hat d\,\, = 0\,\, \Rightarrow \,\,{a_1} - {a_2} = \,\,0\,\,\,......\,\,(ii)$
$[b\,\,c\,\, d]\,\, = \,\,0\,\, $
$\Rightarrow \,\,b\,.\,(c \times \hat d)\,\, = \,\,0$
$ \Rightarrow \,\,\left| {\,\begin{array}{*{20}{c}} 0&1&{ - 1} \\ { - 1}&0&1 \\ {{a_1}}&{{a_2}}&{{a_3}} \end{array}\,} \right|\,\,\, =0$
$ \Rightarrow \,\, - 1\,\,( - {a_3} - {a_1})\,\, - \,\,1\,\,( - {a_2})=0$
$ \Rightarrow \,\,\,\,{a_1} + {a_2} + {a_3} = \,\,0\,\,\, $
$\Rightarrow \,\,2{a_1} +{a_3} = \,\,0\,\,,$ {$(ii)$ પરથી }
$\therefore \,\,\, \frac{{{a_1}}}{{0 + 1}}\,\, = \,\,\,\frac{{{a_2}}}{{1 - 0}}\,\, = \,\,\frac{{{a_3}}}{{ - 1 - 1}}$
$\, \Rightarrow \,\,\frac{{{a_1}}}{1}\,\, = \,\,\frac{{{a_2}}}{1}\,\, = \,\frac{{{a_3}}}{{ - 2}}\,\, = \,\,\lambda ,$
$\therefore \,\,\,{a_1} = \,\lambda \,,\,\,{a_2} = \,\,\lambda \,,\,\,{a_3} = \,\, - 2\lambda $
$\therefore \,\,\,{\lambda ^2} + \,\,{\lambda ^2} + \,\,4{\lambda ^2} = \,\,1$ {$(i)$ પરથી } $\therefore \,\,6{\lambda ^2} = \,\,\,1\,\, $
$\Rightarrow \,\,\lambda \,\, = \,\, \pm \,\,\frac{1}{{\sqrt 6 }}\,\,;\,$
$\,\,\therefore \,\,\hat d\,\, = \,\frac{{i\, + \,\,j\,\, - 2k}}{{\sqrt 6 }}$
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