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If \(\vec{\alpha}=3\hat{i}+4\hat{j}+5\hat{k}\) and \(\vec{\beta}=2\hat{i}+\hat{j}-4\hat{k}\) then express \(\vec{\beta}\) in the form, \(\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2\) where \(\vec{\beta}_1\) is parallel to \(\vec{\alpha}\) and \(\vec{\beta}_2\) is perpendicular to \(\vec{\alpha}.\)

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\(\because \vec{\beta}_1\) is parallel to \(\vec{\alpha},\)

\(\Rightarrow \vec{\beta}_1 = \lambda \vec{\alpha}\) where \(\lambda\) is any scalar quantity

The required expression is

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