Vector Algebra

Let A, B, C be three points whose position vectors respectively are

\(\overrightarrow a = \widehat i + 4\widehat j + 3\widehat k\)

\(\overrightarrow b = 2\widehat i + \alpha \widehat j + 4\widehat k,\,\alpha \in R\)

\(\overrightarrow c = 3\widehat i - 2\widehat j + 5\widehat k\)

If \(\alpha\) is the smallest positive integer for which \(\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c\) are noncollinear, then the length of the median, in \(\Delta\)ABC, through A is :