[Coordinate method in space]

Find the angle between the line passing through points $A(-3 ; 0 ; 1)$ and $B(2 ; 1 ;-1)$, and the line passing through points $C(-2 ; 2 ; 0)$ and $D(1 ; 3 ; 2)$.
Let's find the coordinates of the vectors $\overrightarrow{A B}$ and $\overrightarrow{C D}$:

$$
\begin{aligned}
& \overrightarrow{A B}=(2-(-3) ; 1-0 ;-1-1)=(5 ; 1 ;-2) \\
& \overrightarrow{C D}=(1-(-2) ; 3-2 ; 2-0)=(3 ; 1 ; 2)
\end{aligned}
$$

Let $\phi$ be the angle between the vectors $\overrightarrow{A B}$ and $\overrightarrow{C D}$. Then

$$
\cos \phi=\frac{\overrightarrow{A B} \cdot \overrightarrow{C D}}{|\overrightarrow{A B}| \cdot \mid \overrightarrow{C D}} \left\lvert\,=\frac{5 \cdot 3+1 \cdot 1-2 \cdot 2}{\sqrt{5^{2}+1^{2}+2^{2}} \sqrt{3^{2}+1^{2}+2^{2}}}=\frac{12}{\sqrt{30 \cdot 14}}=\frac{6}{\sqrt{105}}=\frac{2 \sqrt{105}}{35}\right.
$$

If $\alpha$ is the angle between the lines $A B$ and $C D$, then

$$
\cos \alpha=|\cos \phi|=\frac{2 \sqrt{105}}{35}
$$

## Answer

$\arccos \frac{2 \sqrt{105}}{35}$