Question: The parametrization \(\vec{r}\left( t \right)=x\left( t \right)\mathbf{i}+y\left( t \right)\mathbf{j}+z\left( t \right)\mathbf{k}\), where

\[\begin{aligned} & x\left( t \right)={{\cos }^{2}}\left( \omega t \right) \\ & y\left( t \right)=\sin \left( \omega t \right)\cos \left( \omega t \right) \\ & z\left( t \right)=\sin \left( \omega t \right) \\ \end{aligned}\]

is a curve that lies on the surface of the unit sphere.

(a) Verify that the curve lies on the surface of a sphere of radius one centered at the origin.

(b) Find the tangent vector to the curve.

(c) Verify that

\[\vec{r}\left( t \right)\cdot \vec{r}'\left( t \right)=0\]

(d) Is the motion uniform?