Is there any conventional notation for the point given by the intersection between two lines? I wonder if there is any conventional notation to denote the point given by the intersection between those two lines (assuming it exists)?
Any help is highly appreciated!
 A: Yes. 


*

*A line joins two points

*A point meets two lines


With projective geometry and homogeneous coordinates, consider two points with coordinates 
$$\begin{align} 
  {\bf P}_1 = & \pmatrix{x_1 \\ y_1 \\ 1} & {\bf P}_2 = & \pmatrix{x_2 \\ y_2 \\ 1}
\end{align}$$
NOTE: A scalar multiple of the above coordinates denote the same location. For example $\lambda \pmatrix{x & y & 1} = \pmatrix{\lambda x & \lambda y & \lambda}$ has still $(x,y)$ as the location of the points.
The line ${\bf L}$ that joins two two points has coefficients (line "coordinates") 
$$ \begin{align} 
  {\bf L} =&  {\bf P}_1 \times {\bf P}_2 \\ 
  \pmatrix{a \\ b \\ c} & = \pmatrix{x_1\\y_1\\1} \times \pmatrix{x_2\\y_2\\1}
\end{align}$$
Where the equation of the line is $ax+by+c=0$, and $\times$ is the vector cross product
Now consider two lines with coefficients 
$$\begin{align} 
  {\bf L}_1 = & \pmatrix{a_1 \\ b_1 \\ c_1} & {\bf L}_2 = & \pmatrix{a_2 \\ b_2 \\ c_2}
\end{align}$$
The point where the two lines meet is has coordinates
$$ \begin{align} 
  {\bf P} =&  {\bf L}_1 \times {\bf L}_2 \\ 
  \pmatrix{x \lambda \\ y \lambda \\ \lambda} & = \pmatrix{a_1\\b_1\\c_1} \times \pmatrix{a_2\\b_2\\c_2}
\end{align}$$
Example
A line $2x-y-1=0$ meets the line $x+2y-2=0$. Find the location of the point.
$$\begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix} \times \pmatrix{1 \\ 2 \\ -2} = \pmatrix{4 \\ 3 \\ 5}$$
The location is thus $\pmatrix{ \frac{4}{5} & \frac{3}{5} }$
