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%%%------------------------------------------------------------------------------------------------------------
\tdplotsetmaincoords{50}{140}
\begin{tikzpicture}[scale=2,tdplot_main_coords]
{
\draw[thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$a$};
\draw[thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$b$};
\draw[thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{$c$};
}
\pgfmathsetmacro{\ax}{2}
\pgfmathsetmacro{\ay}{2}
\pgfmathsetmacro{\az}{1}
\tdplotsetrotatedcoords{20}{40}{00}
{
\draw[thick,color=red,tdplot_rotated_coords,->] (0,0,0)
        -- (.7,0,0) node[anchor=east]{$a'$};
\draw[thick,color=green!50!black,tdplot_rotated_coords,->] (0,0,0)
        -- (0,.7,0) node[anchor=west]{$b'$};
\draw[thick,color=blue,tdplot_rotated_coords,->] (0,0,0)
        -- (0,0,.7) node[anchor=south]{$c'$};
}
\tdplottransformmainrot{\ax}{\ay}{\az}

\end{tikzpicture}
%%------------------------------------------------------------------------------------------------------------