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From: David Arnold <darnold@northcoast.com>
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Subject: Tangent vector
Date: Wed, 26 Sep 2001 14:47:26 -0700
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All,

In the second image in this file, I am disappointed to find that the vector
doesn't look tangent to the path at the tip of the position vector.

input mp-tool;

beginfig(1);

%initialize scale
 numeric u; 6u=3in;

%draw coordinate axes
 drawdblarrow (-1u,0)--(5u,0);
 label.rt(btex $x$ etex, (5u,0));
 drawdblarrow (0,-1u)--(0,3u);
 label.top(btex $y$ etex, (0,3u));

%path
 path p;
 p:=(-1,1)..(1,2)..(5,2.5);
 p:=p scaled u;
 draw p withcolor blue;

%position vector
 drawarrow (0,0)--(1u,2u) withcolor red;
 label.lrt(btex ${\bf r}(t)$ etex, 0.5[(0,0),(1u,2u)]);
 dotlabel.ulft(btex $(x(t),y(t))$ etex, (1u,2u));
endfig;

beginfig(2);

%initialize scale
 numeric u; 6u=3in;

%draw coordinate axes
 drawdblarrow (-1u,0)--(5u,0);
 label.rt(btex $x$ etex, (5u,0));
 drawdblarrow (0,-1u)--(0,3u);
 label.top(btex $y$ etex, (0,3u));

%path
 path p;
 p:=(-1,1)...(1,2)...(5,2.5);

%position vector
 drawarrow (0,0)--(1u,2u) withcolor red;
 label.lrt(btex ${\bf r}(t)$ etex, 0.5[(0,0),(1u,2u)]);
 dotlabel.ulft(btex $(x(t),y(t))$ etex, (1u,2u));

%calculate the velocity vector
 numeric t;
 t:=directiontime (1u,2u) of p;
 pair d;
 d:=direction t of p;
 drawarrow ((0,0)--d) scaled u shifted (1u,2u);

 p:=p scaled u;
 draw p withcolor blue;

endfig;

end.