 Minimum pairwise dot-product sum of three unit vectors

For three unit vectors, the squared Euclidean norm of their sum is $3+2S$, proving the displayed bound; equality holds exactly when $u+v+w=0$. For fixed unit vectors $u,v$ with $u+v\ne 0$, the Cauchy-Schwarz inequality gives the unique minimizing third vector $w=-(u+v)/\Vert u+v\Vert$, and the minimum is $u\cdot v-\sqrt{2+2u\cdot v}$. Three coplanar directions separated by $120$ degrees attain the global bound.