Let the vectors \({\vec{u}}_{1}=\hat{i}+\hat{j}+a\hat{k},{\vec{u}}_{2}=\hat{i}+b\hat{j}+\hat{k}\) and \({\vec{u}}_{3}=c\…

Let the vectors \({\vec{u}}_{1}=\hat{i}+\hat{j}+a\hat{k},{\vec{u}}_{2}=\hat{i}+b\hat{j}+\hat{k}\) and \({\vec{u}}_{3}=c\hat{i}+\hat{j}+\hat{k}\) be coplanar. If the vectors \({\vec{v}}_{1}=(a+b)\hat{i}+c\hat{j}+c\hat{k},\) \({\vec{v}}_{2}=a\hat{i}+(b+c)\hat{j}+a\hat{k}\) and \({\vec{v}}_{3}=b\hat{i}+b\hat{j}+(c+a)\hat{k}\) are also coplanar, then 6(a + b + c) is equal to

[JEE Main 2023, 08 Apr (Shift 2)]