Vector Algebra

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 ____________.