Let the position vectors of the vertices A, B and C of a triangle be 2 i +2 j + k, i +2 j +2 k and 2 i + j +2 k respectively. Let l_1, l_2 and l_3 be…

Let the position vectors of the vertices $\displaystyle \mathrm{A}, \mathrm{B}$ and C of a triangle be $\displaystyle 2 \hat{i}+2 \hat{j}+\hat{k}, \hat{i}+2 \hat{j}+2 \hat{k}$ and $\displaystyle 2 \hat{i}+\hat{j}+2 \hat{k}$ respectively. Let $\displaystyle l_1, l_2$ and $\displaystyle l_3$ be the lengths of perpendiculars drawn from the ortho center of the triangle on the sides AB, BC and CA respectively, then $\displaystyle l_1^2+l_2^2+l_3^2$ equals :