(Solved): Consider the following vector function. \[ r(t)=\cos (t) i+6 t \mathbf{j}+2 \sin (4 t) \mathbf{k} \ ...

Consider the following vector function. \[ r(t)=\cos (t) i+6 t \mathbf{j}+2 \sin (4 t) \mathbf{k} \] Find each of the following. \[ r^{\prime}(t)= \] \[ \begin{array}{c} r^{\prime}(0)= \\ \left|r^{\prime}(0)\right|= \end{array} \] Find the unit tangent vector \( T(t) \) at the point with the given value of the parameter \( t \). \[ \mathbf{r}(t)=\cos (t) \mathbf{i}+6 t \mathbf{j}+2 \sin (4 t) \mathbf{k}_{,} \quad t=0 \]