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