Vector Algebra

The scalar product of vectors \(\overline{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and a unit vector along the sum of vectors \(\bar{b}=2 \hat{i}-4 \hat{j}+5 \hat{k}\) and \(\bar{c}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}\) is one, then the value of \(\lambda\) is