Let A, B, C be three points whose position vectors respectively are a = i +4 j +3 k; b =2 i +α j +4 k, α ∈ R; c =3 i -2 j +5 k If α is the smallest…

Let $\displaystyle A, B, C$ be three points whose position vectors respectively are $$\begin{aligned} & \vec{a}=\hat{i}+4 \hat{j}+3 \hat{k} \\ & \vec{b}=2 \hat{i}+\alpha \hat{j}+4 \hat{k}, \alpha \in \mathbb{R} \\ & \vec{c}=3 \hat{i}-2 \hat{j}+5 \hat{k} \end{aligned} $$ If $\displaystyle \alpha$ is the smallest positive integer for which $\displaystyle \vec{a}, \vec{b}, \vec{c}$ are noncollinear, then the length of the median, in $\displaystyle \triangle A B C$, through A is :