## [ Coordinate method in space ] Vectors (other).

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

![](https://cdn.mathpix.com/cropped/2024_05_06_2c7701213788eaf04894g-42.jpg?height=128&width=1108&top_left_y=1588&top_left_x=475)

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

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

## Answer

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