Trigonometry · Superchallenge problem

Nonzero vectors $\mathbf a$ and $\mathbf b$ in the plane satisfy $|\mathbf a| = 4$. For every real number $t$, $|\mathbf a - t\mathbf b| \ge |\mathbf a - \mathbf b|$; and for every real number $s$, $|\mathbf b - s\mathbf a| \ge \left|\mathbf b - \tfrac12\mathbf a\right|$. Find the angle between $\mathbf a$ and $\mathbf b$, in degrees.