\documentclass{article}
\usepackage{tikz}
\usepackage{verbatim}

\begin{comment}
:Title: Intersecting lines
:Tags: Axes, Basics, Coordinate systems

An example of using the intersection coordinate systems. Both the ``intersection`` and
``perpendicular`` coordinate system are used. The latter is a special case of the former,
but has the shorter and more convenient ``-|`` and ``|-`` syntax.

\end{comment}

\begin{document}

\begin{tikzpicture}[scale=1.5, >=stealth]
    % Zeichne Grid
    %\draw[step=0.1, color=lightgray] (-1,-1) grid(4,3.5);
    % Draw axes
    \draw [<->] (0,3) node (yaxis) [above] {$y$}|- (3,0) node (xaxis) [right] {$x$};

    \draw [color=red](0,0) -- +(2.5,2.5);
     \draw plot[only marks, mark=x, mark options={blue}] coordinates{(2.5,2.5)}
        node[above, color=blue] {Punkt $(a_1,a_2)$};

    \foreach \y in {0,1,2}
        \draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};

    \foreach \x in {0,1,2}
        \draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
    


    %\draw [color=green] (2.4,2.4) -- +(2.6,2.6) node[right] {Punkt $(a_1,a_2)$};
%    \draw [color=blue](0,3) coordinate (a_1) -- (2,2) coordinate (a_2);
    % Draw two intersecting lines
    \draw [color=green, dashed] (-0.1,2.5) -- (2.5,2.5);
    \draw [color=green, dashed] (2.5,-0.1) -- (2.5,2.5);   
        \node[green] at (-0.25,2.5) {$a_2$};
        \node[green] at (2.5,-0.25) {$a_1$};
        \node[right] at (0.05,2.2) {\tiny{2. Koordinate}};
        \node[right] at (1,-0.4) {\tiny{1. Koordinate}};
    % Calculate the intersection of the lines a_1 -- a_2 and b_1 -- b_2
    % and store the coordinate in c.
%    \coordinate (c) at (intersection of a_1--a_2 and b_1--b_2);
    % Draw lines indicating intersection with y and x axis. Here we use
    % the perpendicular coordinate system
%    \draw[dashed] (yaxis |- c) node[left] {$y'$}
%        -| (xaxis -| c) node[below] {$x'$};
    % Draw a dot to indicate intersection point
%    \fill[red] (c) circle (2pt);
\end{tikzpicture}
\end{document}