Let v → = α i ^ + 2 j ^ - 3 k ^ ,   w → = 2 α i ^ + j ^ - k ^ , and u → be…

We have,

$\left[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}\right]=\overrightarrow{u}·\left(\overrightarrow{v}\times \overrightarrow{w}\right)$

Now,

$\min \left(\left[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}\right]\right)=-\alpha \sqrt{3401}$

$\Rightarrow -\left|\overrightarrow{u}\right|\left|\overrightarrow{v}\times \overrightarrow{w}\right|=-\alpha \sqrt{3401}$

$\Rightarrow \alpha \left|\overrightarrow{v}\times \overrightarrow{w}\right|=\alpha \sqrt{3401}$

$\Rightarrow \left|\overrightarrow{v}\times \overrightarrow{w}\right|=\sqrt{3401}$

$\Rightarrow \left|\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ \alpha & 2& -3\\ 2\alpha & 1& -1\end{array}\right|\right|=\sqrt{3401}$

$\Rightarrow \left|\hat{i}-5\alpha \hat{j}-3\alpha \hat{k}\right|=\sqrt{3401}$

$\Rightarrow \sqrt{1+25{\alpha }^2+9{\alpha }^2}=\sqrt{3401}$

$\Rightarrow 34{\alpha }^2=3400$

$\Rightarrow {\alpha }^2=100$

$\Rightarrow \alpha =10$        (as $\alpha >0$)

So,

$\overrightarrow{u}=\lambda \left(\hat{i}-5\alpha \hat{j}-3\alpha \hat{k}\right)$

$\Rightarrow {\alpha }^2={\lambda }^2\left(1+25{\alpha }^2+9{\alpha }^2\right)$

$\Rightarrow 100={\lambda }^2\left(1+34\times 100\right)$

$\Rightarrow {\lambda }^2=\frac{100}{3401}$

And,

${\left|\overrightarrow{u}·\hat{i}\right|}^2={\lambda }^2=\frac{100}{3401}=\frac{m}{n}$

Hence, $m+n=3501$