How many lines intersect two lines and a point in $\mathbb{P}^3$

Question

I am trying to understand some basic intersection theory in $\mathbb{P}^3$, the complex projective space of lines in $\mathbb{C}^4$. Suppose that $l_1,l_2$ be two lines and that $p$ be a point in $\mathbb{P}^3$. The question is if there must exist a line $l$ in in $\mathbb{P}^3$ that intersects $l_1,l_2,p$. This post says that there exists a unique line, but leaves out details. I can't fill in the details.

Answer 1

If $l_1, l_2$ are skew (i.e. disjoint) and $p$ is on neither of these lines, then the unique line through $p$ and cutting both $l_1,l_2$ is the line $l=\overline {p,q}$ obtained by joining $p$ to the intersection point $$q=l_2\cap (p*l_1)$$ of $l_2$ with the unique plane $p*l_1$ containing $p$ and $l_1$.

Edit: an example
At Mike's request I'll follow the procedure in the case he suggests, where: $$l_1=\{x_1=x_2=0\},l_2=\{x_3=x_4=0\}, p=[a,b,c,d] \; \operatorname {with} a\operatorname {or} b\neq0\operatorname {and} c\operatorname {or} d\neq0$$ (the inequalities at the end ensuring that $p\notin l_1 \cup l_2$)
The plane $p*l_1$ has as equation $-bx_1+ax_2=0$ and it cuts $l_2$ at the point $q=[a,b,0,0] $.
Finally the requested line $l=\overline {p,q}$ has as parametric representation $up+vq=[ua+va,ub+vb,uc,ud]$ (with $u,v\in \mathbb C$ not both zero) or, changing parameters, $$l=\overline {p,q}=\{[ta,tb,uc,ud]\vert [t,u]\in \mathbb P^1(\mathbb C)\} $$ For $t=0,u=1$ you get the point $[0,0,c,d]$ on $l_1$, for $t=1,u=0$ you get the point $[a,b,0,0]$ on $l_2$, and finally for $[t=1,u=1]$ you get $p=[a,b,c,d]$. Et voilà !