Vector Algebra

A unit vector $\hat{\mathbf{e}}=a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$ is coplanar with the vectors $\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$, and $3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-5 \hat{\mathbf{k}}$. If $\hat{\mathbf{e}}$ is perpendicular to the vector $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$, then $2 a^{2}+3 b^{2}+4 c^{2}=$