 Solution

The triangle inequality for the two unit vectors gives $0<\Vert u+v\Vert \le 2$. Taking reciprocals and using the negative sign in $\lambda =-1/\Vert u+v\Vert$ yieldsIn fact equality would require $u=v$, by the equality condition in the Cauchy-Schwarz inequality. Since the two vectors here are linearly independent, the inequality is strict for every admissible pair.