Let \( (\alpha, \beta, \gamma) \) be the co-ordinates of the foot of the perpendicular drawn from the point \( (5, 4, 2) \) on the line \( \vec{r} = (\hat{-i} + 3\hat{j} + \hat{k}) + \lambda (2\hat{i} + 3\hat{j} - \hat{k}) \). Then the length of the projection of the vector \( \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) on the vector \( 6\hat{i} + 2\hat{j} + 3\hat{k} \) is:
