= Solution
Take the initial <velocity> perpendicular to a uniform <magnetic field>. The magnetic <Lorentz force> is perpendicular to the motion, has constant magnitude $|q|vB$, and supplies the <centripetal force>:
$$
\frac{mv^2}{r}=|q|vB,\qquad\boxed{r=\frac{mv}{|q|B},\quad\omega_c=\frac vr=\frac{|q|B}{m}.}
$$
The rotation sense is fixed by $q\mathbf v\times\mathbf B$. A component of <velocity> parallel to the field remains constant and produces a <helical motion in a uniform magnetic field>; a circle requires that component to vanish. The speed-independent <cyclotron frequency> here assumes nonrelativistic motion.
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