Example 1. Find the torsion of the helical line

$$
r=a \cos t i + a \sin t j + h t k
$$
Solution. We find the derivatives of the given vector

$$
\begin{aligned}
& \frac{d \mathbf{r}}{d t}=-a \sin t \mathbf{i}+a \cos t \mathbf{j}+h \mathbf{k} \\
& \frac{d^{2} \mathbf{r}}{d t^{2}}=-a \cos t \mathbf{i}-a \sin t \mathbf{j} \\
& \frac{d^{3} \mathbf{r}}{d t^{3}}=a \sin t \mathbf{i}-a \cos t \mathbf{j}
\end{aligned}
$$

The mixed product of these vectors is

$$
\left(\frac{d \mathbf{r}}{d t}, \frac{d^{2} \mathbf{r}}{d t^{2}}, \frac{d^{3} \mathbf{r}}{d t^{3}}\right)=\left|\begin{array}{ccc}
-a \sin t & a \cos t & h \\
-a \cos t & -a \sin t & 0 \\
a \sin t & -a \cos t & 0
\end{array}\right|=a^{2} h
$$

In example I. §5, it was found that

$$
\left|\left[\frac{d \mathbf{r}}{d t}, \frac{d^{2} \mathbf{r}}{d t^{2}}\right]\right|^{2}=a^{2}\left(a^{2}+h^{2}\right)
$$

Using formula (1), we obtain for the torsion

$$
\frac{1}{T}=\frac{h}{a^{2}+h^{2}}
$$

Thus, the torsion of the helical line is the same at all its points.