If a =2 i - j + k, b = i + j -2 k and c = i +3 j - k and the projection of vector c +λ b on vector a is 2 √ 6, then find the value of λ.

If $\displaystyle \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\displaystyle \overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and the projection of vector $\displaystyle \vec{\mathrm{c}}+\lambda \vec{\mathrm{b}}$ on vector $\displaystyle \vec{\mathrm{a}}$ is $\displaystyle 2 \sqrt{6}$, then find the value of $\displaystyle \lambda$.