For a Lipschitzian vector field in $\R^n$, angular velocity of its trajectories with respect to any stationary point is bounded by the Lipschitz constant. The same is true for a rotation speed around any integral submanifold of the field. However, easy examples show that a trajectory of a $C^{\infty}$-vector field in $\R^3$ can make in finite time an infinite number of turns around a straight line. We show that for a trajectory of a polynomial vector field in $\R^3$, its rotation rate around any algebraic curve is bounded in terms of the degree of the curve and the degree and size of the vector field. As a consequence, we obtain a linear in time bound on the number of intersections of the trajectory with any algebraic surface.