$=\left\{\frac{(3 \hat{j}+4 \hat{k}) \cdot(\hat{i}+\hat{j})}{\sqrt{2}}\right\}\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$
$=-\frac{3(\hat{i}+\hat{j})}{\sqrt{2} \times \sqrt{2}}=\frac{3(\hat{i}+\hat{j})}{2}$
$\overrightarrow{\mathrm{b}_{1}}+\overrightarrow{\mathrm{b}_{2}}=\overrightarrow{\mathrm{b}}$
$\overrightarrow{b_{2}}=\vec{b}-\overrightarrow{b_{1}}$
$ = \left( {\left. {3\hat j + 4\hat k - \frac{3}{2}} \right)(\hat i + \hat j)} \right.$
$\boxed{\overrightarrow {{b_2}} = - \frac{3}{2}\widehat {\text{i}} + \frac{3}{2}\widehat {\text{j}} + 4\widehat {\text{k}}}$
$\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} = \begin{array}{*{20}{c}}
{\widehat i}&{\widehat j}&{\widehat k}\\
{\frac{3}{2}}&{\frac{3}{2}}&0\\
{ - \frac{3}{2}}&{\frac{3}{2}}&4
\end{array}$
$\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} = \widehat {\rm{i}}(6) - \widehat {\rm{j}}({\rm{6}}) + \widehat {\rm{k}}\left( { - \frac{9}{4} + \frac{9}{4}} \right)$
$\Rightarrow 6 \hat{i}-6 \hat{j}+\frac{9}{2} \hat{k}$
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