Solution
= Solution
When $\mathbf x\cdot\mathbf y=\cos(2\pi/3)=-1/2$, the squared <norm> of their sum is $2+2(-1/2)=1$. Therefore
$$
\boxed{\lambda=-1,\qquad\mathbf z_*=-\mathbf x-\mathbf y,\qquad S_{\min}=-\frac12-1=-\frac32.}
$$
The 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.