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\…

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

[JEE Main 2023, 24 Jan (Shift 2)]