Answer
$${\bf{T}}\left( t \right) = \left\langle {\frac{1}{{\sqrt {14} }},\frac{2}{{\sqrt {14} }},\frac{3}{{\sqrt {14} }}} \right\rangle {\text{ and }}\kappa \left( t \right) = 0$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \left\langle {2t + 1,4t - 5,6t + 12} \right\rangle \cr
& {\text{Calculate }}{\bf{v}}\left( t \right) \cr
& {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr
& {\bf{v}}\left( t \right) = \frac{d}{{dt}}\left\langle {2t + 1,4t - 5,6t + 12} \right\rangle \cr
& {\bf{v}}\left( t \right) = \left\langle {2,4,6} \right\rangle \cr
& {\text{Find the unit tangent vector }}{\bf{T}} \cr
& {\bf{T}}\left( t \right) = \frac{{{\bf{v}}\left( t \right)}}{{\left| {{\bf{v}}\left( t \right)} \right|}} \cr
& {\bf{T}}\left( t \right) = \frac{{\left\langle {2,4,6} \right\rangle }}{{\left| {\left\langle {2,4,6} \right\rangle } \right|}} = \frac{{\left\langle {2,4,6} \right\rangle }}{{\sqrt {4 + 16 + 36} }} \cr
& {\bf{T}}\left( t \right) = \frac{{\left\langle {2,4,6} \right\rangle }}{{\sqrt {56} }} \cr
& {\bf{T}}\left( t \right) = \left\langle {\frac{2}{{\sqrt {56} }},\frac{4}{{\sqrt {56} }},\frac{6}{{\sqrt {56} }}} \right\rangle \cr
& {\bf{T}}\left( t \right) = \left\langle {\frac{2}{{2\sqrt {14} }},\frac{4}{{2\sqrt {14} }},\frac{6}{{2\sqrt {14} }}} \right\rangle \cr
& {\bf{T}}\left( t \right) = \left\langle {\frac{1}{{\sqrt {14} }},\frac{2}{{\sqrt {14} }},\frac{3}{{\sqrt {14} }}} \right\rangle \cr
& {\text{Therefore}}{\text{, the curvature is}} \cr
& \kappa \left( t \right) = \frac{1}{{\left| {\bf{v}} \right|}}\left| {\frac{{d{\bf{T}}}}{{dt}}} \right| \cr
& \frac{{d{\bf{T}}}}{{dt}} = \frac{d}{{dt}}\left\langle {\frac{1}{{\sqrt {14} }},\frac{2}{{\sqrt {14} }},\frac{3}{{\sqrt {14} }}} \right\rangle \cr
& \frac{{d{\bf{T}}}}{{dt}} = 0 \cr
& \kappa \left( t \right) = \frac{1}{{\left| {\bf{v}} \right|}}\left| 0 \right| \cr
& \kappa \left( t \right) = 0 \cr
& \cr
& {\bf{T}}\left( t \right) = \left\langle {\frac{1}{{\sqrt {14} }},\frac{2}{{\sqrt {14} }},\frac{3}{{\sqrt {14} }}} \right\rangle {\text{ and }}\kappa \left( t \right) = 0 \cr} $$
