\(\vec{a}=2\hat{i}-\hat{j}+3\hat{k},\overset{⇀}{b}=3\hat{i}-5\hat{j}+\hat{k}If\vec{a}\times \vec{c}=\vec{c}\times \vec{b…

\(\vec{a}\times \vec{c}=\vec{c}\times \vec{b}\\ (\vec{a}+\vec{b})\times \vec{c}=0\\ \vec{c}=\lambda (\vec{a}+\vec{b})=\lambda (5\hat{l}-6\hat{ı}+4\hat{k})\\ |\vec{c}{|}^{2}={\lambda }^{2}(77)\\ (\vec{a}+\vec{c})\cdot (\vec{c}+\vec{b})=168\\ \Rightarrow \vec{a}\cdot \vec{c}+\vec{a}\cdot \vec{b}+|\vec{c}{|}^{2}+\vec{c}\cdot \vec{b}=168\\ 14+\vec{c}\cdot (\vec{a}+\vec{b})+|\vec{c}{|}^{2}=168\\ 14+c|\vec{a}+{\left.\vec{b}\right|}^{2}+|\vec{c}{|}^{2}=168\\ 14+77\lambda +77{\lambda }^{2}=168\\ {\lambda }^{2}+\lambda -2=0\\ \lambda =-2or\lambda =1\\ |\overset{⇀}{c}{|}^{2}=77{\lambda }^{2}=308or77\)