Let a line L passing through the point (1,1,1) be perpendicular to both the vectors 2 i +2 j + k and i +2 j +2 k. If P ( a, b, c ) is the foot of…

Let a line L passing through the point $\displaystyle (1,1,1)$ be perpendicular to both the vectors $\displaystyle 2 \hat{i}+2 \hat{j}+\hat{k}$ and $\displaystyle \hat{i}+2 \hat{j}+2 \hat{k}$. If $\displaystyle \mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})$ is the foot of perpendicular from the origin on the line L, then the value of $\displaystyle 34(\mathrm{a}+\mathrm{b}+\mathrm{c})$ is :