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


\begin{document}

\begin{tikzpicture}[scale=1.5, on grid]

%\foreach \x in {0,...,4}
%    \foreach \y in {0,...,3}
%        \foreach \z in {0,...,5}
%            \fill[ball color=gray] (\x,\y,-\z) circle (0.05);

%\begin{scope}[every node/.append style={yslant=0},yslant=0]
% \draw [help lines](0,0) grid (5,5);
%\end{scope}

%\begin{scope}[every node/.append style={yslant=-1},yslant=-1]
% \draw [help lines](0,0) grid (5,5);
%\end{scope}

%\begin{tikzpicture}[scale=3]

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

    %\draw[help lines] (0,0) grid (3,3);
    %\draw[help lines] (0,0,-1) grid (3,3,-1);
    %\draw[help lines] (0,0,-2) grid (3,3,-2);
    
    
%initial    
    \draw[-triangle 45, color=blue] (0,0,0) -- (4,0,0) node[midway,sloped,above] {$\vec a$}; %x
    \draw[-triangle 45,style=dashed, color=red] (0,0,0) -- (1,1,-1) node[midway,sloped,above] {$\vec b$}; %z
    \draw[-triangle 45, color=orange] (0,0,0) -- (1,3,0) node[midway,sloped,above] {$\vec c$}; %y

%vorn

    \draw[-, color=blue] (1,3,0) -- (5,3,0);
    \draw[-, color=red] (4,0,0) -- (5,1,-1) ;
    \draw[-, color=orange] (4,0,0) -- (5,3,0);
%untere Vebindung im z (hinten)
    \draw[-, color=blue, style=dashed] (1,1,-1) -- (5,1,-1);

%vec b oben 
    \draw[-, color=red] (1,3,0) -- (2,4,-1) ;
    \draw[-, color=red] (5,3,0) -- (6,4,-1) ;

%vec a oben / hinten
     \draw[-, color=blue] (2,4,-1) -- (6,4,-1) ;

%vec c beide hinteren
     \draw[-, color=orange, style=dashed] (1,1,-1) -- (2,4,-1) ;
     \draw[-, color=orange] (5,1,-1) -- (6,4,-1);

    %\draw[-, color=blue] (1,3) -- (4,3) node[midway,sloped,above] {};
    %\draw[-, color=red] (3,0,0) -- (4,1,-2) node[midway,sloped,above] {};
    %\draw[-, style=dashed, color=blue] (1,1,-2) -- (4,1,-2) node[midway,sloped,above] {};


    %\draw[-, color=red] (1,3,-2) -- (2,4,-2) node[midway,sloped,above] {};

%    \draw[style=dashed]  (2,2) -- (6,2);
%    \draw[style=dashed]  (4,0) -- (6,2);


%    \draw (0,0) +(0:1cm) arc (0:45:1cm);

 %   \path (0,0) ++(22.5:.75cm) node{$\alpha$};

\end{tikzpicture}
\end{document} 