$ \text{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}. ; \text{If } \alpha \text{ is the smallest positive integer for which } \vec{a}, \vec{b}, \vec{c} \text{ are non-collinear, then the length of the median in } \triangle ABC \text{ through A is :}$