Given : $$\overrightarrow a = (\alpha ,1, - 1)$$ and $$\overrightarrow b = (2,1, - \alpha )$$

$$\overrightarrow c = \overrightarrow a \times \overrightarrow b = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr \alpha & 1 & { - 1} \cr 2 & 1 & { - \alpha } \cr } } \right|$$

$$ = ( - \alpha + 1)\widehat i + ({\alpha ^2} - 2)\widehat j + (\alpha - 2)\widehat k$$

Projection of $$\overrightarrow c $$ on $$\overrightarrow d = - \widehat i + 2\widehat j - 2\widehat k$$

$$ = \left| {\overrightarrow c \,.\,{{\overrightarrow d } \over {|d|}}} \right| = 30$$ {Given}

$$ \Rightarrow \, = \left| {{{\alpha - 1 - 4 + 2{\alpha ^2} - 2\alpha + 4} \over {\sqrt {1 + 4 + 4} }}} \right| = 30$$

On solving $$\alpha = {{ - 13} \over 2}$$ (Rejected as $$\alpha > 0$$)

and $$\alpha = 7$$