A particle follows the path r(t)=⟨t2,ln(t),2t⟩\mathbf{r}(t) = \langle t^2, \ln(t), 2t \rangler(t)=⟨t2,ln(t),2t⟩ for t>0t > 0t>0. Find the unit tangent vector T(t)\mathbf{T}(t)T(t) at t=1t=1t=1.
⟨23,13,23⟩\langle \frac{2}{3}, \frac{1}{3}, \frac{2}{3} \rangle⟨32,31,32⟩
⟨13,23,23⟩\langle \frac{1}{3}, \frac{2}{3}, \frac{2}{3} \rangle⟨31,32,32⟩
⟨1,1,2⟩\langle 1, 1, 2 \rangle⟨1,1,2⟩
⟨25,15,45⟩\langle \frac{2}{5}, \frac{1}{5}, \frac{4}{5} \rangle⟨52,51,54⟩