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

$\vec{v} \times \vec{w}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ \alpha & 2 & -3 \\ 2 \alpha & 1 & -1\end{array}\right|=\hat{i}-5 \alpha \hat{j}-3 \alpha \hat{k}$ <br/><br/>$$\left[ {\matrix{ {\overrightarrow u } & {\overrightarrow v } & {\overrightarrow w } \cr } } \right] = \overrightarrow u .\left( {\overrightarrow v \times \overrightarrow w } \right)$$ <br/><br/>$=|\vec{u}||\vec{v} \times \vec{w}| \times \cos \theta$ <br/><br/>$=\alpha \sqrt{34 \alpha^{2}+1} \cos \theta$ <br/><br/>$[\vec{u} \vec{v} \vec{w}]_{\min }=-\alpha \sqrt{3401}$ <br/><br/>$\alpha \sqrt{34 \alpha^{2}+1} \times(-1)=-\alpha \sqrt{3401}$ <br/><br/>(taking $\cos \theta=1$ ) <br/><br/>$\Rightarrow \alpha=10$ <br/><br/>$\vec{v} \times \vec{w}=\hat{i}-50 \hat{j}-30 \hat{k}$ <br/><br/>$\cos \theta=-1 \Rightarrow \vec{u}$ is antiparallel to $\vec{v} \times \vec{w}$ <br/><br/>$\vec{u}=-|\vec{u}| \cdot \frac{\vec{v} \times \vec{w}}{|\vec{v} \times \vec{w}|}=\frac{-10(\hat{i}-50 \hat{j}-30 \hat{k})}{\sqrt{3401}}$ <br/><br/>$|\vec{u} \cdot \hat{i}|^{2}=\left|\frac{-10}{\sqrt{3401}}\right|^{2}=\frac{100}{3401}=\frac{m}{n}$ <br/><br/>$m+n=3501$