Let \( \vec{\alpha} = 4\hat{i} + 3\hat{j} + 5\hat{k} \) and \( \vec{\beta} = \hat{i} + 2\hat{j} - 4\hat{k} \). Let \( \vec{\beta}_1 \) be parallel to \( \vec{\alpha} \) and \( \vec{\beta}_2 \) be perpendicular to \( \vec{\alpha} \).
If \( \vec{\beta} = \vec{\beta}_1 + \vec{\beta}_2 \), then the value of \( 5\vec{\beta}_2 \cdot (\hat{i}, \hat{j}, \hat{k}) \) is:
