可以保存切点和向量终点的坐标，然后利用 perpendicular 坐标系统即可画出。
\documentclass[tikz,border=5pt]{standalone}
\usetikzlibrary{decorations.markings}
\makeatletter
\def\my@tangentparse#1{\in@{ at}{#1}%
\ifin@\my@tangentparse@#1\@nil\else\my@tangentparse@ last at#1\@nil\fi}
\def\my@tangentparse@#1 at#2\@nil{\pgfkeysalso{postaction=decorate,
decoration = { % 对path进行装饰（decorate）
markings, % 启用marking
mark = at position #2 % 在指定位置#1添加marking
with{ % marking的具体内容
% 定义(0, 0)为tangent point xxx，即切点
\coordinate (tangent point #1) at (0, 0);
% 定义(1, 0)为tangent unit vector xxx，即单位切向量
\coordinate (tangent unit vector #1) at (1, 0);
% 定义(0, 1)为tangent orthogonal unit vector xxx，即单位法向量
\coordinate (tangent orthogonal unit vector #1) at (0, 1);
}
},
}}
\tikzset{
tangent/.code = \my@tangentparse{#1},
use tangent/.style = { % 定义use tangent样式，参数#1是切点的序号
shift = (tangent point #1), % 以tangent point-xxx为原点
x = (tangent unit vector #1), % 以tangent unit vector-xxx为x轴单位向量
y = (tangent orthogonal unit vector #1)% 以tangent orthogonal unit vector-xxx为y轴单位向量
},
use tangent/.default = last,
}
\makeatother
\begin{document}
\begin{tikzpicture}
\draw[->] (-1,0) -- (3,0) node[anchor=south east]{$x$};
\draw[->] (0,-1) -- (0,2) node[anchor=south west]{$y$};
\draw[tangent=A at 0.65,tangent=B at 0.8] (0, 0) .. controls (1, 0) and (1, 1) .. (2, 0.5);
\fill[use tangent=A] circle(1pt) node[below=1] {$p$};
\draw[black!20,use tangent=A] (-1.5, 0) -- (1.5, 0);%切线
\draw[->, use tangent=A] (0, 0) -- (0, 1) coordinate(L) node[scale=0.4,above=2] {$vector$}; %法线
\draw[dotted,black!10, shorten <=-1cm] (L) -- (0,0 |- L); %辅助线
\draw[dotted,black!10] (L) -- (0,0 -| L); %辅助线
\draw[red,->] (tangent point A) -- (L |- tangent point A);
\draw[red,->] (tangent point A) -- (L -| tangent point A);
\draw[->, use tangent=B] (0, 0) -- (0, 1) coordinate(L);
\draw[red,->] (tangent point B) -- (L |- tangent point B);
\draw[red,->] (tangent point B) -- (L -| tangent point B);
\end{tikzpicture}
\end{document}
问 如何精确的绘制二维向量的两个分向量？