Question: Let \(\vec{u}\left( t \right)=\left( -\sqrt{t}\sin t,t,{{t}^{2/3}} \right)\) and \(\vec{v}\left( t \right)=\left( -\sqrt{t}\sin t,{{\cos }^{2}}t,-{{t}^{1/3}} \right)\), compute

\[\frac{d}{dt}\left\langle \vec{u}\left( t \right),\vec{v}\left( t \right) \right\rangle \]

and

\[\frac{d}{dt}\left\langle \vec{u}\left( t \right),\vec{u}\left( t \right) \right\rangle \]