Assertion : Let a =3 i - j , b =2 i + j -3 k . If b = b _ 1 + b _ 2 such that b _ 1 is…

Assertion : Let \(\vec{a}=3 \hat{i}-\hat{j},\) \(\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}} .\)If \(\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{b}}_{1}+\overrightarrow{\mathrm{b}}_{2}\) such that \(\mathrm{b}_{1}\) is collinear with \(\stackrel{\rightharpoonup}{\mathrm{a}}\) and \(\overrightarrow{\mathrm{b}}_{2}\)is perpendicular to \(\stackrel{\rightharpoonup}{\mathrm{a}}\)is possible, then \(\overrightarrow{\mathrm{b}}_{2}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-3 \hat{\mathrm{k}} .\)
Reason : If \(\stackrel{\rightharpoonup}{\mathrm{a}}\) and \(\overrightarrow{\mathrm{b}}\) are non–zero, non–collinear vectors, then \(\overrightarrow{\mathrm{b}}\) can be expressed as \(\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{b}}_{1}+\overrightarrow{\mathrm{b}}_{2}\), where \(\overrightarrow{\mathrm{b}}_{1}\) is collinear with \(\stackrel{\rightharpoonup}{\mathrm{a}}\)and \(\overrightarrow{\mathrm{b}}_{2}\) is perpendicular to \(\stackrel{\rightharpoonup}{\mathrm{a}}\).