Chapter 14.3, Problem 31E

Trajectories on circles and spheres Determine whether the following trajectories lie on a circle in ${ℝ}^2$ or sphere in ${ℝ}^3$ centered at the origin. If so, find the radius of the circle or sphere and show that the position vector and the velocity vector are everywhere orthogonal.

27. $r(t)=\langle \sin t+\sqrt{3}\cos t,\sqrt{3}\sin t-\cos t\rangle$, for 0 ≤ t ≤ 2π