Answer
$$\mathbf{T} =\left(-\frac{2}{3} \sin t\right) \mathbf{i}+\left(\frac{2}{3} \cos t\right) \mathbf{j}+\frac{\sqrt{5}}{3} \mathbf{k}$$
$$\text { Length } =3\pi $$
Work Step by Step
Since
$\mathbf{r}=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{j}+\sqrt{5} t \mathbf{k} $
Then
$ \mathbf{v}=(-2 \sin t) \mathbf{i}+(2 \cos t) \mathbf{j}+\sqrt{5} \mathbf{k} $ and
\begin{align*}
|\mathbf{v}|&=\sqrt{(-2 \sin t)^{2}+(2 \cos t)^{2}+(\sqrt{5})^{2}} \\
&=3
\end{align*}
Hence
\begin{align*}
\mathbf{T}&=\frac{\mathbf{v}}{|\mathbf{v}|}\\
&=\left(-\frac{2}{3} \sin t\right) \mathbf{i}+\left(\frac{2}{3} \cos t\right) \mathbf{j}+\frac{\sqrt{5}}{3} \mathbf{k} \\
\text { Length }&=\int_{0}^{\pi}|\mathbf{v}| d t\\
&=\int_{0}^{\pi} 3 d t\\
&=3t \bigg|_{0}^{\pi}\\
&=3\pi
\end{align*}
