documentclass{article}
usepackage{pgfplots}
usepackage{tikz-3dplot}
pgfplotsset{compat=1.3}

begin{document}
tdplotsetmaincoords{60}{120}

begin{figure}
centering
begin{tikzpicture}[scale=1.5, tdplot_main_coords, axis/.style={->,blue,thick},
vector/.style={-stealth,red,very thick},
vector guide/.style={dashed,red,thick}]
begin{axis}[
axis lines=center,
axis on top,
xlabel={$x$}, ylabel={$y$}, zlabel={$t$},
domain=0:1,
y domain=0:2*pi,
xmin=-1.5, xmax=1.5,
ymin=-1.5, ymax=1.5, zmin=0.0,
every axis x label/.style={at={(rel axis cs:0,0.5,0)},anchor=south},
every axis y label/.style={at={(rel axis cs:0.5,0,0)},anchor=north},
every axis z label/.style={at={(rel axis cs:0.5,0.5,0.9)},anchor=west},
samples=30]
addplot3 [surf, colormap/blackwhite, shader=flat] ({x*cos(deg(y))},{x*sin(deg(y))},{x});
%standard tikz coordinate definition using x, y, z coords
coordinate (O) at (0,0,0);

%tikz-3dplot coordinate definition using x, y, z coords

pgfmathsetmacro{axone}{5}
pgfmathsetmacro{ayone}{3}
pgfmathsetmacro{azone}{4}
pgfmathsetmacro{axtwo}{3}
pgfmathsetmacro{aytwo}{5}
pgfmathsetmacro{aztwo}{0}

coordinate (P1) at (axone,ayone,azone);
coordinate (P2) at (axtwo,aytwo,aztwo);

draw[vector] (O) -- (P1);
draw[vector] (O) -- (P2);

end{axis}
end{tikzpicture}
end{figure}
end{document}

documentclass[tikz,border=3.14mm]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=1.16}

begin{document}
begin{tikzpicture}[scale=1.5,axis/.style={->,blue,thick},
vector/.style={-stealth,red,very thick},
vector guide/.style={dashed,red,thick}]
begin{axis}[
axis lines=center,
axis on top,
xlabel={$x$}, ylabel={$y$}, zlabel={$t$},
domain=0:1,
y domain=0:2*pi,
xmin=-1.5, xmax=1.5,
ymin=-1.5, ymax=1.5, zmin=0.0,
every axis x label/.style={at={(rel axis cs:0,0.5,0)},anchor=south},
every axis y label/.style={at={(rel axis cs:0.5,0,0)},anchor=north},
every axis z label/.style={at={(rel axis cs:0.5,0.5,0.9)},anchor=west},
samples=30]
addplot3 [surf, colormap/blackwhite, shader=flat] ({x*cos(deg(y))},{x*sin(deg(y))},{x});
%standard tikz coordinate definition using x, y, z coords
coordinate (O) at (0,0,0);

%tikz-3dplot coordinate definition using x, y, z coords

pgfmathsetmacro{axone}{0.5}
pgfmathsetmacro{ayone}{0.3}
pgfmathsetmacro{azone}{0.4}
pgfmathsetmacro{axtwo}{0.3}
pgfmathsetmacro{aytwo}{0.5}
pgfmathsetmacro{aztwo}{0}

coordinate (P1) at (axone,ayone,azone);
coordinate (P2) at (axtwo,aytwo,aztwo);

draw[vector] (O) -- (P1);
draw[vector] (O) -- (P2);

end{axis}
end{tikzpicture}
end{document}