= Solution
For the stated angle, $q=\cos(2\pi/3)=-1/2$, so $\|u+v\|=\sqrt{2+2q}=1$. The minimizing <unit vector> is therefore $w=-(u+v)$, and
$$
\boxed{\lambda=-1,\qquad S_{\min}=-\frac12-1=-\frac32.}
$$
Indeed all three pairwise <dot products> equal $-1/2$, so the vectors lie in one plane, separated by $120$ degrees. This is also the global <minimum pairwise dot-product sum of three unit vectors>: for any three unit vectors,
$$
0\le\|u+v+w\|^2=3+2S,
$$
so $S\ge-3/2$, with equality exactly when $u+v+w=0$.
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