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}+\…

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 vector \(\vec{v}_1=(a+b) \hat{i}+c \hat{j}+c \hat{k}, \vec{v}_2=a \hat{i}+(b+c) \hat{j}+a \hat{k}\) 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, 8 Apr (Shift 2)]