the problem lies with your eigenvectors for 3, as both are orthogonal to (-1,1,1). so apply gram-schmidt to just those 2.

for example if we take u1 = v1 = (1,1,0), then gram-schmidt gives:

u2 = v2 - [(u1.v1)/(u1.u1)]u1 = (1,0,1) - (1/2)(1,1,0) = (1/2,-1/2,1).

by construction, u2 is orthogonal to u1, and (1/2,-1/2,1).(-1,1,1) = -1/2 + -1/2 + 1 = 0.