Vector Algebra

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 \(6(\mathrm{a}+\mathrm{b}+\mathrm{c})\) is equal to :