Describing the motion of a particle (sphere) If I have the following position at time t : $\hat{r}(t) = 3\cos(t)\hat{i} + 4\cos(t)\hat{j} + 5\sin(t)\hat{k}$ , then how can I tell if the particle's path lies on a sphere or not? If e.g. the second term was simply $t$ and not a trigonometric function, I know that it would be a spiral, since $t$ will change along (in this case) y-axis on this curve, but if it was a constant then we would simply have an ellipse; but I don't know how to explain or derive a sphere from the given position vector, I tried to use spherical coordinates but to no avail. Could someone explain how one can see if the path of the particle lies on the sphere? If it indeed is a sphere, then from the norm of the position we know that the radius will be 5 units.

Let $t = \theta + \frac{\pi}{2}$, so that $\hat{r}(t)$ gives $x = 3\cos(\theta + \frac{\pi}{2}) = -3\sin\theta, y= -4\sin \theta, z = 5\cos\theta$.
Comparing to the spherical coordinates: $x = r\sin\theta \cos\phi, y = r\sin\theta \sin\phi, z = r\cos\theta$.
Then radius of sphere $r = 5$, but we need to find the fixed azimuth $\phi$ such that $5\cos \phi = -3$, and $5 \sin\phi = -4$.
