Let the vectors \({\vec{u}}_{1}=\overset{^}{i}+\overset{^}{j}+a\overset{^}{k},{\vec{u}}_{2}=\overset{^}{i}+b\overset{^}{…

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

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