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

Given $\overrightarrow{a}=\alpha \widehat{i}+\widehat{j}-\widehat{k}\&\overrightarrow{b}=2\widehat{i}+\widehat{j}-\alpha \widehat{k}$

$\overrightarrow{a}\times \overrightarrow{b}=\left(1-\alpha \right)\widehat{i}+\left({\alpha }^2-2\right)\widehat{j}+\left(\alpha -2\right)\widehat{k}$

Projection of $\overrightarrow{a}\times \overrightarrow{b}$ on $-\widehat{i}+2\widehat{j}-2\widehat{k}$

$\Rightarrow \frac{\left(\overrightarrow{a}\times \overrightarrow{b}\right).\left(-\widehat{i}+2\widehat{j}-2\widehat{k}\right)}{3}=30$

$\Rightarrow 2{\alpha }^2-\alpha -91=0$

$\Rightarrow \alpha =7,\frac{-13}{2}$