Let v =α i +2 j -3 k, w =2 α i + j - k and u be a vector such that | u |=α>0. If the minimum value of the scalar triple product [ u v w ] is -α √…

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