Let $\overline{v}=\alpha \widehat{i}+2\widehat{j}-3\widehat{k},\overline{w}=2\alpha \widehat{i}+\widehat{j}-\widehat{k}$  and  $\overline{u}$  be a vector such that $\left|\widehat{u}\right|=a>0$ . If the minimum value of the scalar triple product $\left[\overline{u}\overline{v}\overline{w}\right]$   is $-\alpha \sqrt{3401}$   and ${|\overline{u}\cdot \widehat{i}|}^2=\frac{m}{n}$  , where m and n are coprime natural numbers, then  is equal to ________

We have  $v=\alpha \widehat{i}+2\widehat{j}-3\widehat{k},w=2\alpha \widehat{i}+\widehat{j}-\widehat{k}\left|\widehat{u}\right|=\alpha$
Now,  $[uvw=u.(v\times w)]=\left|u\right||v\times w|\cos \theta =\alpha |v\times w|\cos \theta$
For minimum value of  $\left[uvw\right],\theta =\pi$
$\therefore {\left[uvw\right]}_{\min }=-\alpha [v\times w]$  $\Rightarrow |v\times w|=\sqrt{3401}$  
Now,  $v\times w=\left|\begin{array}{l}ijk\\ \alpha 2-3\\ 2\alpha 1-1\end{array}\right|$ $=i-5\alpha j-3\alpha k$ 
Now,   $|v\times w|=\sqrt{3401} \Rightarrow \alpha =10[\because \alpha >0] \therefore v\times w=i-50j-30k$
 
Again,   $u=10(-\frac{v\times w}{|v\times w|}) =\frac{-10}{\sqrt{3401}}(i-50j-30k)$
Now,  ${|u.i|}^2={\left(\frac{-10}{\sqrt{3401}}\right)}^2=\frac{100}{3401}$  $\therefore m=100,n=3401$
 . So,   $m+n=3501$