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.

The locus is a sphere of radius 5 centered at origin. Since y/x is constant, there is a cutting plane though the poles. You are on meridians of longitude 180*arctan (4/3) and another on the other side of sphere.
