Vector Algebra

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