Answer
$\langle \frac{\sqrt 7}{5},\frac{3}{5},\frac{3}{5}\rangle $
Work Step by Step
$\textbf{r}(t) = \langle f(t), g(t), h(t)\rangle$
$\textbf{r}'(t) = \langle f'(t), g'(t), h'(t)\rangle$
$\textbf{r}(t) = \langle \sqrt 7e^t,3e^t,3e^t\rangle$
$\textbf{r}'(t) = \langle \sqrt 7e^t,3e^t,3e^t\rangle$
$|\textbf{r}'(t)| = \sqrt {(\sqrt 7e^t)^2 + (3e^t)^2 + (3e^t)^2} = \sqrt {25e^{2t}}$
$\textbf{T}(t) = \frac{\textbf{r}'(t)}{|\textbf{r}'(t)|} = \frac{\langle \sqrt 7e^t,3e^t,3e^t\rangle}{ \sqrt {25e^{2t}}} = 5e^t$
$\textbf{T}(ln(2)) = \frac{\langle \sqrt 7e^{ln(2)},3e^{ln(2)},3e^{ln(2)}\rangle}{ 5e^{ln(2)}} = \langle \frac{\sqrt 7}{5},\frac{3}{5},\frac{3}{5}\rangle $
