Let \( A, B, C \) be three points whose position vectors respectively are
\( \vec{a}=\hat{i}+4\hat{j}+3\hat{k} \)
\( \vec{b}=2\hat{i}+\alpha \hat{j}+4\hat{k}, \ \alpha \in \mathbb{R} \)
\( \vec{c}=3\hat{i}-2\hat{j}+5\hat{k} \)
If \( \alpha \) is the smallest positive integer for which \( \vec{a}, \vec{b}, \vec{c} \) are non-collinear, then the length of the median, in \( \Delta ABC \), through \( A \) is:
