Equation of line in 3D geometry

Question

Consider the following equations of two lines, $l_1$ and $l_2$. It is given that these two lines do not intersect.
$\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$
These are the equation of a pair of lines.
On doing the derivation of this equation for one line(say, $l_1$), we know that $\vec{r}$ is the position vector of an arbitrary point (say, $P$) on $l_1$
But the same $\vec{r}$ is also present in the equation of $l_2$, that means that point $P$ is present on both the lines, but it has been given that the two lines do not intersect. Please tell me what is the logic behind representing both lines with $\vec{r}$?