[ Coordinate method in space ] [ Parametric equations of a line]

Given points $A(1 ; 0 ; 1), B(-2 ; 2 ; 1), C(2 ; 0 ; 3)$ and $D(0 ; 4 ;-2)$. Formulate the parametric equations of the line passing through the origin and intersecting the lines $A B$ and $C D$.
Through the point $O(0 ; 0 ; 0)$ and the line $A B$, we draw the plane $\alpha$. Let $\overrightarrow{n_{\mathbf{I}}}=(a ; b ; c)$ be a non-zero vector perpendicular to the plane $\alpha$. Then $\overrightarrow{n_{\mathbf{1}}} \cdot \overrightarrow{O A}=0$ and $\overrightarrow{n_{\mathbf{1}}} \cdot \overrightarrow{O B}=0$, or

$$
\left\{\begin{array}{l}
a+c=0 \\
-2 a+2 b+c=0
\end{array}\right.
$$

Let $a=2$. Then $c=-2, b=\frac{1}{2}(2 a-c)=3$. Therefore, the equation of the plane $\alpha$ is $2 x+3 y-2 z=0$. Through the point $O(0 ; 0 ; 0)$ and the line $C D$, we draw the plane $\beta$. Let $\overrightarrow{n_{2}}=(p ; q ; r)$ be a non-zero vector perpendicular to the plane $\beta$. Then $\overrightarrow{n_{2}} \cdot \overrightarrow{O C}=0$ and $\overrightarrow{n_{2}} \cdot \overrightarrow{O D}=0$, or

$$
\left\{\begin{aligned}
2 p+3 r & =0 \\
4 q-2 r & =0
\end{aligned}\right.
$$

Let $p=3$. Then $r=-2, q=\frac{\mathbf{1}}{2} r=-1$. Therefore, the equation of the plane $\beta$ is $3 x-y-2 z=0$. The desired line is the intersection of the planes

$$
2 x+3 y-2 z=0 \text{ and } 3 x-y-2 z=0
$$

We will find its parametric equations. For this, let $x=8 t$ and solve for $y$ and $z$ the system of equations

$$
\left\{\begin{array}{l}
3 y-2 z=-16 t \\
y+2 z=24 t
\end{array}\right.
$$

We get that $y=2 t, z=11 t$. Thus, the parametric equations of the desired line $l$ are

$$
x=8 t, y=2 t, z=11 t \text{. }
$$

The line $l$ lies in the same plane as the line $A B$ and is not parallel to the line $A B$, since the direction vector of the line $l$ (its coordinates $-(-8 ; 2 ; 11)$) is not collinear with the vector $\overrightarrow{A B}=(-3 ; 2 ; 0)$. Therefore, the lines $l$ and $A B$ intersect. Similarly, we check that the lines $l$ and $C D$ also intersect.

## Answer

$x=8 t, y=2 t, z=11 t$.