Let $\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$. If the projection of $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ on the vector $-\hat{i}+2 \hat{j}-2 \hat{k}$ is 30, then $\alpha$ is equal to :

<p>Given : $\overrightarrow a = (\alpha ,1, - 1)$ and $\overrightarrow b = (2,1, - \alpha )$</p> <p>$$\overrightarrow c = \overrightarrow a \times \overrightarrow b = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr \alpha & 1 & { - 1} \cr 2 & 1 & { - \alpha } \cr } } \right|$$</p> <p>$$ = ( - \alpha + 1)\widehat i + ({\alpha ^2} - 2)\widehat j + (\alpha - 2)\widehat k$$</p> <p>Projection of $\overrightarrow c$ on $\overrightarrow d = - \widehat i + 2\widehat j - 2\widehat k$</p> <p>$$ = \left| {\overrightarrow c \,.\,{{\overrightarrow d } \over {|d|}}} \right| = 30$$ {Given}</p> <p>$$ \Rightarrow \, = \left| {{{\alpha - 1 - 4 + 2{\alpha ^2} - 2\alpha + 4} \over {\sqrt {1 + 4 + 4} }}} \right| = 30$$</p> <p>On solving $\alpha = {{ - 13} \over 2}$ (Rejected as $\alpha > 0$)</p> <p>and $\alpha = 7$</p>