Answer
\begin{align*}
\mathbf{T}&=\frac{1}{\sqrt{3}} \mathbf{i}-\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}\\
\text{Length }& =3 \sqrt{3}
\end{align*}
Work Step by Step
Since
$\mathbf{r}=(2+t) \mathbf{i}-(t+1) \mathbf{j}+t \mathbf{k} $
Then
$\mathbf{v}=\mathbf{i}-\mathbf{j}+\mathbf{k} \Rightarrow|\mathbf{v}|=\sqrt{1^{2}+(-1)^{2}+1^{2}}=\sqrt{3}$
Hence
\begin{align*}
\mathbf{T}&=\frac{\mathbf{v}}{|\mathbf{v}|}\\
&=\frac{1}{\sqrt{3}} \mathbf{i}-\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}\\
\text{Length }&=\int_{0}^{3} \sqrt{3} d t\\
&=[\sqrt{3} t]_{0}^{3}\\
&=3 \sqrt{3}
\end{align*}
