As $\vec{a} \times(\vec{b} \times \vec{c})=\vec{b}+\lambda \vec{c}$
$\Rightarrow \overrightarrow{ a } \cdot \overrightarrow{ c }(\overrightarrow{ b })-(\overrightarrow{ a } \cdot \overrightarrow{ b }) \overrightarrow{ c }=\overrightarrow{ b }+\lambda \overrightarrow{ c }$
$\Rightarrow \vec{a} \cdot \vec{c}=1, \vec{a} \cdot \vec{b}=-\lambda$
$\Rightarrow(3 \hat{ i }+\hat{ j }) \cdot(\hat{ i }+2 \hat{ j }+\hat{ k })=-\lambda$
$\Rightarrow \lambda=-5$
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