I have a random vector $v$ whose expected squared projection length along a unit vector $\hat{u}$ is large, while the expected squared length is small, i.e. $\mathbf{E}[|<v,\hat{u}>|^2]$ is large and $\mathbf{E}[||v||^2]$ is small. Does this imply that the angle between $v$ and $\hat{u}$ is small?
Without the probablistic assumption, if the projection of a vector along a direction is large and its length is small, then the cosine of the angle is large, which in turn imply that the angle is small. Does this remain true in the probabilitic setting?