Difficult# Cauchy-Schwarz Equality Condition

LINALG-ZYXVE4

The Cauchy-Schwarz inequality gives a lower bound for the product $\lVert \vec v\rVert\lVert \vec w\rVert$ for any pair of vectors $\vec v,\vec w$.

Which condition on $\vec v,\vec w$ ensures that this lower bound is *sharp* (i.e., that the inequality is an equality)?