Answer
$\mathbf{T}(t)=\frac{\mathbf{r}^{\prime}(t)}{\left\|\mathbf{r}^{\prime}(t)\right\|}, \quad \mathbf{r}(t)=\frac{\mathbf{T}(t)}{\left\|\mathbf{T}^{\prime}(t)\right\|}, \quad \mathbf{B}(t)=\mathbf{T}(t) \times \mathbf{N}(t)$
Work Step by Step
Consider $\mathbf{r}(t)$ is smooth vector valued function:
\[
\begin{aligned}
\frac{\mathbf{r}^{\prime}(t)}{\left\|\mathbf{r}^{\prime}(t)\right\|} =\mathbf{T}(t) &\\
\frac{\mathbf{T}(t)}{\left\|\mathbf{T}^{\prime}(t)\right\|} =\mathbf{r}(t) &\\
& \mathbf{T}(t) \times \mathbf{N}(t)=\frac{\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)}{\left\|\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)\right\|}=\mathbf{B}(t) \\
\end{aligned}
\]
