 Solution

When $\mathbf{x}\cdot \mathbf{y}=\cos (2\pi /3)=-1/2$, the squared norm of their sum is $2+2(-1/2)=1$. ThereforeThe minimizing vector is indeed a unit vector. All three pairwise scalar products then equal $-1/2$, so these three unit vectors lie in a plane, separated by $120$ degrees.