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.

In your case spherical coordinates should work I think? with $\sin (a) = \frac{3}{5}$ ,$$r=5(\sin (a)\cos (t), \cos (a)\cos (t), \sin (t))$$ which is indeed a spherical coordinate with $a$ is the angle around $z$ axis and $t$ is the angle away from $z$ axis. Hope this could help. I think one can try to show $$\frac{d}{dt}\left( (r_x-a)^2+(r_y-b)^2+(r_z-c)^2\right)=0$$ with (a,b,c) can be derived by set $t=t_0,t_1,t_2,t_4$ and use this 4 points to find the sphere across them and (a,b,c) is the center of the sphere. Hope this can help.
