Answer
$${\bf{T}}\left( t \right) = \left\langle {\cos t, - \sin t} \right\rangle {\text{ and }}{\bf{N}}\left( t \right) = \left\langle { - \sin t, - \cos t} \right\rangle $$
Work Step by Step
$$\eqalign{ & {\bf{r}}\left( t \right) = \left\langle {2\sin t,2\cos t} \right\rangle \cr & {\text{Calculate }}{\bf{v}}\left( t \right){\text{ and }}\left| {{\bf{v}}\left( t \right)} \right| \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = \frac{d}{{dt}}\left\langle {2\sin t,2\cos t} \right\rangle \cr & {\bf{v}}\left( t \right) = \left\langle {2\cos t, - 2\sin t} \right\rangle \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {4{{\cos }^2}t + 4{{\sin }^2}t} \cr & \left| {{\bf{v}}\left( t \right)} \right| = 2 \cr & \cr & {\text{Find the unit tangent vector }}{\bf{T}}\left( t \right) \cr & {\bf{T}}\left( t \right) = \frac{{{\bf{v}}\left( t \right)}}{{\left| {{\bf{v}}\left( t \right)} \right|}} = \frac{{\left\langle {2\cos t, - 2\sin t} \right\rangle }}{2} \cr & {\bf{T}}\left( t \right) = \left\langle {\cos t, - \sin t} \right\rangle \cr & \cr & {\text{Find the principal unit normal vector }}{\bf{N}}\left( t \right) \cr & {\bf{N}}\left( t \right) = \frac{{{\bf{T}}'\left( t \right)}}{{\left| {{\bf{T}}'\left( t \right)} \right|}} \cr & {\bf{T}}'\left( t \right) = \left\langle { - \sin t, - \cos t} \right\rangle \cr & \left| {{\bf{T}}'\left( t \right)} \right| = \sqrt {{{\sin }^2}t + {{\cos }^2}t} = 1 \cr & {\text{Then}} \cr & {\bf{N}}\left( t \right) = \frac{{\left\langle { - \sin t, - \cos t} \right\rangle }}{1} \cr & {\bf{N}}\left( t \right) = \left\langle { - \sin t, - \cos t} \right\rangle \cr} $$
