If \(\bar{a}+2 \bar{b}+3 \bar{c}=\overline{0}\) and \((\bar{b} \times \bar{b})+(\bar{b} \times \bar{c})+(\bar{c} \times \bar{a})=\lambda(\bar{b} \times \bar{c})\), then \(\lambda\) has the value

\(\Rightarrow(\vec{a}+\vec{b}+\vec{c}) \times(\vec{a}+2 \vec{b}+3 \vec{c})=\overrightarrow{0} \)
\( \Rightarrow 2 \vec{a} \times \vec{b}+3 \vec{a} \times \vec{c}+\vec{b} \times \vec{a}+3 \vec{b} \times \vec{c}+\vec{c} \times \vec{a}~+\) \(2 \vec{c} \times \vec{b}=\overrightarrow{0} \)
\( \Rightarrow \vec{a} \times \vec{b}+\vec{b} \times \vec{c}-2 \vec{c} \times \vec{a}=\overrightarrow{0} \)
\( \Rightarrow \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=3 \vec{c} \times \vec{a}=3 \vec{c} \times(-2 \vec{b}-3\)\(\vec{c}) \quad[\text { from (1)] } \)
\( \Rightarrow \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=-6 \vec{c} \times \vec{b} \)
\( \Rightarrow \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=6 \vec{b} \times \vec{c} \)
\( \Rightarrow \lambda=6\)