\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{snakes}
\usepackage{verbatim}

\tikzstyle{mybox} = [draw=red, fill=blue!20, very thick,
    rectangle, rounded corners, inner sep=10pt, inner ysep=20pt]

\begin{document}

\begin{tikzpicture}[scale=1.5]
%\begin{tikzpicture}[scale=3]

    %\draw[step=0.1, color=lightgray] (-1,-1) grid(4.5,4.5);

    \draw[help lines] (-5,-5) grid (5,5);
    %\draw[-triangle 45] (0,0) -- (4,0) node[right] {$x$};

    \draw [-triangle 45] (0,0) -- (xyz cs:x=4) node[right] {$x$};
    \foreach \x/\xtext in {0,1,2,3/a_1}
      \draw[xshift=\x cm, thick] (0pt,1pt) -- (0pt,-1pt) node[below]
            {$\xtext$};

    \draw [-triangle 45] (0,0) -- (xyz cs:y=4); % node[anchor=north] {$y$};
        \foreach \y/\ytext in {1,2,3/a_2}
      \draw[yshift=\y cm, thick] (1pt,0pt) -- (-1pt, 0pt) node[left]
            {$\ytext$};


    \draw [-triangle 45] (0,0) -- (xyz cs:z=4);%node[anchor = north] {$z$}
    \foreach \z/\zt in {1,2/a_3,3}
    % \draw [shift=\z cm, thick] {0.1, 0, \z} --  {-0.1, 0, \z};
%      \draw[shift=\z cm, thick] (0pt,1pt) -- (0pt,-1pt) node[left] {$z$};
    \draw [thick] (0.025pt, 0, \z) --  (-0.025pt, 0, \z) node[left] {$\zt$};

    \draw [color=orange, thick, -triangle 45](0,0,0) -- (3,3,2) node[sloped,above] {Vektor ("Pfeil")  $\vec a$} ;

%http://www.statistiker-wg.de/pgf/tutorials.htm
\path %(.42,-.767) node[below] {$x$}
        (0,4.05) node[above] {$y$}
       % (-1.5,-1.653) node[left] {$z$ (3. Koordinate)};
       (0, 0,4.2) node {$z$ (3. Koordinate)};

        \draw [color=red, dashed, thin] (0,0,2) -- (3,0,2);
        \draw [color=red, dashed, thin] (0,0,2) -- (0,3,2);
        \draw [color=red, dashed, thin] (0,3,2) -- (3,3,2);

        \draw [color=blue, dashed, thin] (3,0,0) -- (3,0,2);
        \draw [color=blue, dashed, thin] (3,0,2) -- (3,3,2);
        \draw [color=blue, dashed, thin] (3,0,0) -- (3,3,0);
        \draw [color=blue, dashed, thin] (3,3,0) -- (3,3,2);


        \draw [color=green, dashed, thin] (0,3,0) -- (0,3,2);
        \draw [color=green, dashed, thin] (0,3,2) -- (3,3,2);
        \draw [color=green, dashed, thin] (0,3,0) -- (3,3,0);

        %\draw (3.1,3,2.1) node[right, text width=15mm]{Punkt $\left ( a_1, a_2, a_3 \right )$};

        %\fill[ball color=gray] (3,3,2) circle (0.05); % 3D lighting effect
%\fill[white] (0,0) circle (\R); % just a white circle
%\draw (0,0) circle (0.05);

 %   \draw[-triangle 45] (0,0) -- (0,4) node[right] {$z$};
 %    \foreach \z in {1,2,3}
 %      \draw (\z cm,1pt) -- (\z cm,-1pt) node[anchor=north] {$\z$};
\end{tikzpicture}
\end{document} 