How to simply rotate without any shifts

Question

So I am trying to draw a simple vector at the (-2,-2) point that is exactly 60 degrees. However, when I rotate, it places the arrow somewhere completely different. I do not know what else to do because putting in xshift or yshift options are too much guesswork involved. What do I do?

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,calc,positioning,decorations.pathreplacing,calligraphy}
\usepackage{tikzsymbols}

\begin{document}
\begin{tikzpicture}
\filldraw (-2,-2) circle (1.5pt);
\draw[thick] (-2,-2) ..controls +(60:1) and +(160:1) ..(0,-1)
            ..controls +(340:1) and +(250:1) ..(2.5,0);
\draw (-2.25,-2.25) rectangle (-1.75,-1.75);
\draw[-stealth,rotate=60,thick] (-2,-2)--(-1.5,-2);
\end{tikzpicture}

\end{document}

Answer 1

By default, rotations are done around the origin point (0,0), to rotate around another point, it must be explicitly specified.

I quote the manual:

/tikz/rotate around={<degree>:<coordinate>} (no default)

Rotates the coordinate system by <degree> around the point <coordinate>.

\draw[-stealth,rotate around ={60:(-2,-2)},thick,blue] (-2,-2)node{x}--(-1.5,-2);

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,calc,positioning,decorations.pathreplacing,calligraphy}
\usepackage{tikzsymbols}

\begin{document}
\begin{tikzpicture}
\filldraw (-2,-2) circle (1.5pt);
\draw[thick] (-2,-2) ..controls +(60:1) and +(160:1) ..(0,-1)
            ..controls +(340:1) and +(250:1) ..(2.5,0);
\draw (-2.25,-2.25) rectangle (-1.75,-1.75);
\draw[-stealth,rotate around ={60:(-2,-2)},thick,blue] (-2,-2)node{x}--(-1.5,-2);
\end{tikzpicture}

\end{document}

Answer 2

To long to be a comment ...

Do you looking for something like this:

Arrows is drawn by using polar coordinate:

\documentclass[tikz,border=3mm]{standalone}

\begin{document}
    \begin{tikzpicture}
\filldraw (-2,-2) circle (1.5pt);
\draw[thick] (-2,-2) ..controls +(60:1) and +(160:1) ..(0,-1)
            ..controls +(340:1) and +(250:1) ..(2.5,0);
\draw (-2.25,-2.25) rectangle (-1.75,-1.75);
\draw[-stealth] (-2,-2) -- ++ (60:1);
    \end{tikzpicture}
\end{document}