pgfplot: 3D version of a plane and its the gradient vector [closed]

Question

Checking pgfplot for the addplot3-command was easy! I've found this very nice possibility to plot a plane:

\begin{tikzpicture}
\begin{axis}
\addplot3[
    surf,
    mesh,
]
{x+y+1};
\end{axis}
\end{tikzpicture}

But to emphasize the meaning of the gradient vector I would like to add a vector (1,1,0). Furthermore I need labels for my coordinate system. And how can I add points, e.g. "the origin", to indicate that the functions returns 1 for x=y=0? Thanks for advice.

Here is a specification of the underlying problem: adding a circle or vector refers to the invisible 2d-coordinate system but not to the one plotted.

\begin{tikzpicture}
\begin{axis}
\addplot3[
    surf,
        mesh,
]
{x+y+1};
\end{axis}
\draw[->] (1,1,0) -- (2,2,0);
\draw (0,0,1) circle (10pt);
\end{tikzpicture}

\begin{center}
\begin{tikzpicture}
\begin{axis}[
    xlabel={$x$},
    ylabel={$y$}]
\addplot3[
    surf, mesh
        ]
{x+y+1};
\draw[->, ultra thick] (0,0,0) -- (1,1,0);
\fill[red] (0,0,0) circle (2pt);
\draw[->] (1,1,0) -- (2,2,0);
\end{axis}
\end{tikzpicture}
\end{center}

Of course I just mixed up the environment of the addplot3 with the axis environment and possible other subsequent elements. Everythings fine now.