\documentclass{standalone}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{amsmath}

\tikzset{>=latex}
\tikzstyle{proj}=[blue!70!black,very thin,dashed]
\tikzstyle{axis}=[->,thick]
\tikzstyle{vector}=[-stealth,axis,blue!80!black]
\tikzstyle{unit vector}=[->,green!35!black,very thick]

\begin{document}

\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords]

    \pgfmathsetmacro{\coordlen}{7}
    \pgfmathsetmacro{\px}{3}
    \pgfmathsetmacro{\py}{4}
    \pgfmathsetmacro{\pr}{5}
    \pgfmathsetmacro{\pz}{3.75}
    \pgfmathsetmacro{\prr}{6.25}

    \coordinate (O) at (0,0,0);
    \coordinate (P) at (\px,\py,\pz);
    \coordinate (Pxy) at (\px,\py,0);
    \coordinate (Pxz) at (\px,0,\pz);
    \coordinate (Pyz) at (0,\py,\pz);
    \coordinate (Px) at (\px,0,0);
    \coordinate (Py) at (0,\py,0);
    \coordinate (Pz) at (0,0,\pz);

    \draw (O) node[anchor=south east]{$O$};

    \draw[axis] (O) -- (\coordlen,0,0);
    \draw[axis] (O) -- (0,\coordlen,0);
    \draw[axis] (O) -- (0,0,\coordlen);

    \draw[vector] (O) -- (P) node[anchor=south east]{$\boldsymbol{r}$};

    \draw[proj] (O) -- (3.75,5,0);
    \tdplotdrawarc[proj]{(0,0,\pz)}{\pr}{0}{360}{proj}{}
    
    \tdplotgetpolarcoords{\px}{\py}{\pz}
    \tdplotsetthetaplanecoords{\tdplotresphi}

    \draw[tdplot_rotated_coords,proj] (\prr,0,0) arc (0:90:\prr);

    \draw ($(O)!.5!(P)$) node [anchor=north]{$r$};
    \tdplotdrawarc[tdplot_rotated_coords,->]{(O)}{.5}{0}{\tdplotrestheta}{anchor=south west}{$\vartheta$}
    \tdplotdrawarc[->]{(O)}{.5}{0}{\tdplotresphi}{anchor=north}{$\varphi$}

    \tdplotsetrotatedcoords{\tdplotresphi}{\tdplotrestheta}{0}
    \tdplotsetrotatedcoordsorigin{(P)}

    \draw[tdplot_rotated_coords,unit vector] (0,0,0) -- (1,0,0) node[anchor=north west]{$\boldsymbol{e}_{\vartheta}$};
    \draw[tdplot_rotated_coords,unit vector] (0,0,0) -- (0,1,0) node[anchor=west]{$\boldsymbol{e}_{\varphi}$};
    \draw[tdplot_rotated_coords,unit vector] (0,0,0) -- (0,0,1) node[anchor=south]{$\boldsymbol{e}_r$};

\end{tikzpicture}

\end{document}