How to draw tangent vectors and component vectors on a curve

Question

I am searching for a way to have a figure drawn using LaTeX. I am aware of some packages TikZ and PSTricks that could accomplish this, and have been tinkering around with them recently but do not have enough skill set at the moment to accomplish this drawing. Is there a way someone can accomplish the drawing, preferably with TikZ but could be any package your most comfortable with in a reduced amount of coding (non-advanced) to where I could understand what is happening from the code to the picture. If this is not possible, then I will have to study the package more and work with it until I understand the method (code) to create the graphic. The drawing is below.

Note: This is a rough sketch. The optimal output would have smooth quadratic curves and all quantities are vectors.

Answer 1

To give you a correctly answer, I need to add a code to put together all the next parts . The trick here: we place a node on the path with option sloped so we a get a tangent. To show this tangent line, we place the node with anchor=south west. The corner of the bow is now correctly placed,and finally we give correct size to the node.

All the parts together:

\documentclass[]{scrartcl}  
\usepackage{tikz}   
\usetikzlibrary{arrows}       
\begin{document}

 \begin{tikzpicture}[allow upside down]
\draw[red,line width=1pt] (0,0) .. controls ++(1,6) and ++(-1,2) .. (10,4)
      node[sloped,inner sep=0cm,above,pos=.5,
      anchor=south west,
      minimum height=(10.5)*0.3cm,minimum width=(10.5)*.3cm](N){};

\path (N.south west)%
           edge[-stealth',blue] node[left] {$\vec{ n}$} (N.north west)
           edge[-stealth',blue] node[above] {$\vec{ t}$} (N.south east);
\draw[-stealth',gray]  (N.south west)  --%
      node[below] {$\vec{t_a}$} (N.south west -| N.south east); 
\draw[-stealth',gray]  (N.south west)  --%
      node[right] {$\vec{t_o}$} (N.south west |- N.south east);

  \end{tikzpicture}    
 \end{document}

Complement about tangents and normals

\documentclass[11pt]{scrartcl}   
\usepackage{tikz}       
 \usetikzlibrary{arrows}      
\begin{document}

 \begin{tikzpicture}[allow upside down]
    \draw[red,line width=1pt] (0,0)
    .. controls +(right:6cm) and +(left:4cm) .. (2,6)
    \foreach \p in {0,5,...,100} {
      node[sloped,inner sep=0cm,above,pos=\p*0.01,
      anchor=south west,
      minimum height=(10+\p)*0.03cm,minimum width=(10+\p)*0.03cm]
      (N \p){}
    }
    .. controls +(right:2cm) and +(right:3cm) .. (6,2)
    \foreach \p in {5,10,...,100} {
      node[sloped,inner sep=0cm,above,pos=\p*0.01,
      anchor=south west,
      minimum height=(110-\p)*0.03cm,minimum width=(110-\p)*0.03cm]
      (N2 \p){}
    }
    ;
    \foreach \p in {0,5,...,100} {
      \draw[-latex,blue] (N \p.south west) -- (N \p.north west);
      \draw[-latex,color=green!50!black] (N \p.south west) -- (N \p.south east);
    }
    \foreach \p in {5,10,...,100} {
      \draw[-latex,blue] (N2 \p.south west) -- (N2 \p.north west);
      \draw[-latex,color=green!50!black] (N2 \p.south west) -- (N2 \p.south east);
    }
  \end{tikzpicture}
\vspace{2cm}
\begin{tikzpicture}[>=latex']
\coordinate (A) at (1,1);
\coordinate  (B) at (3,4);
\draw
    [->]
    (A) --  node [above] {\(\vec v\)}(B)  
    ; 
    \draw[red,->] (A)-- node [above] {\(\vec a\)}(A|-B) ; 
    \draw[red,->] (A|-B)--node [left] {\(\vec b\)}(B) ;
\end{tikzpicture}

 \end{document}

Next a code to show how to draw component vector (it's possible to create a macro to do this. Instead of \draw [->](A) -- (B) node [above,midway] {\(\vec v\)};you can write \draw [->](A) -- node [above] {\(\vec v\)}(B);.

With pgf 2.1 CVS now it's possible to use a new option edge label or edge node to place directly the label (see the last picture)

\documentclass[11pt]{scrartcl}   
\usepackage{tikz}       
 \usetikzlibrary{arrows}      
\begin{document}

\begin{tikzpicture}[>=latex']
  \coordinate (A) at (1,1);
\coordinate  (B) at (3,4);     
    \draw [->](A) --  node [above] {\(\vec v\)}(B);
    \path (A)--(A|-B) coordinate (C) ;
    \draw[red,->] (A) -- node[left] {\(\vec a\)} (C) ; 
    \draw[red,->] (C)-- node[above] {\(\vec b\)}(B) ;  
\end{tikzpicture} 

 \end{document}

Only with PGF 2.10 CVS

Answer 2

change the function to whatever you need:

\documentclass{article}
\usepackage{pstricks-add}
\begin{document}

\begin{pspicture}(-1,-1)(9,6)
\psaxes[ticksize=0 5pt]{->}(9,6)
\psset{algebraic,linewidth=0.8pt,arrowscale=2} \def\function{-x*(x-8)/3}
\psplot[linewidth=1.5pt,algebraic]{0}{8}{\function}
\psplotTangent*[linecolor=red,arrows=->]{6}{-2.5}{\function}
\uput{10pt}[45](*5 {\function}){$\vec{v}$}
\pcline[linestyle=dashed,linecolor=blue]{->}(*6 {\function})(*6 {\function-1})
\naput[npos=0.95]{$\vec{a_y}$}
\pcline[linestyle=dashed,linecolor=blue]{->}(*6 {\function-1})(4.5,0|*6 {\function-1})
\naput{$\vec{a_x}$}
\pcline[linestyle=dashed,linecolor=blue]{->}(*6 {\function})(4.5,0|*6 {\function-1})
\nbput{$\vec{a}$}
\pcline[linestyle=dashed,linecolor=green]{->}(4.5,0|*6 {\function-1})(4.5,0|*5 {\function+0.5})
\naput{$\vec{\omega}$}
\end{pspicture}

\end{document}

and with the latest pstricks-add.tex from http://texnik.dante.de/tex/generic/pstricks-add/:

\documentclass{article}
\usepackage{pstricks-add}
\begin{document}

\psset{unit=4}
\begin{pspicture}(1,-1)(4,1)
\pscurve[showpoints=true](2.1,-0.2)(2.5,0.2)(3.2,0.4)(3.8,-0.2)
\psTangentLine[Tnormal,arrows=->,linecolor=red](2.5,0.2)(3.2,0.4)(3.8,-0.2){3.5}{0.5}
\psTangentLine[arrows=->,linecolor=blue](2.5,0.2)(3.2,0.4)(3.8,-0.2){3.5}{0.5}
\pcline[linestyle=dashed]{->}(OCurve)(ETangent|OCurve)\naput{$v_x$}
\pcline[linestyle=dashed]{->}(ETangent|OCurve)(ETangent)\naput{$v_y$}
\end{pspicture}

\end{document}

Answer 3

The TikZ manual provides some excellent tutorials. The first two or three of them can be done in a matter of hours and will teach you enough of the basics so that you should be able to solve your problem.

To give some examples about the "trickier" parts of your drawing: You can create an arrow with a label like this:

\draw
    [->] % Add arrow tip at the end
    (0,0) -- (3,1) % Start/end coordinate
    node [above, midway] {\(\vec a\)} % Add a label centered on the arrow just drawn
    ;

To find out how to draw dashed lines, search the manual for "dashed". The manual is excellent, and simply searching it will answer many questions you come up with. Drawing the curve is possible with Bezier curves, they are briefly explained in the first tutorial.

That should be all you need. (And using TikZ gets better, only the first few pictures feel like swimming through honey. And then it becomes quite a lot of fun sometimes.)