We have $b \times c = \,|b||c|\sin \alpha \,k = \sin \alpha \,k$, where $k$ is a unit vector perpendicular to $b$ and $c.$
$ \Rightarrow \,|b \times c|\, = \sin \alpha \Rightarrow k = \frac{{b \times c}}{{|b \times c|}}$
Any vector $a$ can be written as a linear combination of $i,\,\,j$ and $k.$
Let $a = {a_1}i + {a_2}j + {a_3}k.$
Now $a\,.\,b = a\,.\,i = {a_1},$ $a\,.\,c = a\,.\,j = {a_2}$
and $a\,.\frac{{b \times c}}{{|b \times c|}} = a\,.\,k = {a_3}$
Thus $(a\,.\,b)b + (a\,.\,c)c + \frac{{a\,.\,(b \times c)}}{{|b \times c|}}(b \times c|$
$ = {a_1}b + {a_2}c + {a_3}\frac{{(b \times c)}}{{|b \times c|}} = {a_1}i + {a_2}j + {a_3}k = a$.
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