\documentclass{article}
\usepackage{mpgraphics}
\begin{ltxpreamble}
\usepackage{amsmath,amssymb}
\end{ltxpreamble}
\begin{document}
\begin{mpdisplay}
a=70pt; b=70pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(67,7));
label(btex $z$ etex,(7,67));
label(btex $0$ etex,(1,-6));
label(btex $P(a,b,c)$ etex,(48,36));
draw (30,30)..(30,-20);
draw (30,30)..(0,50) dashed evenly;
draw (0,0)..(30,-20) dashed evenly;
draw (30,-20)..(-20,-20) dashed evenly;
draw (30,-20)..(50,0) dashed evenly;
label(btex $a$ etex,(-26,-20));
label(btex $b$ etex,(50,6));
label(btex $c$ etex,(-6,50));
pickup pencircle scaled 0.1pt;
draw (6,-4)..(6,3); draw (6,3)..(0,7);
draw (-13,-20)..(-8,-15); draw (-8,-15)..(-15,-15);
draw (43,0)..(37.5,-5); draw (37.5,-5)..(45,-5);
draw (30,-13)..(24,-9); draw (24,-9)..(24,-16);
pickup pencircle scaled 3.5pt;
drawdot (30,30);
\end{mpdisplay}

\begin{mpdisplay}
a=70pt; b=70pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
path xz, yz, xy;
xz = buildcycle(z0..(0,60),(0,60)..(-30,30),(-30,30)..(-30,-30),(-30,-30)..z0);
yz = buildcycle(z0..(60,0),(60,0)..(60,60),(60,60)..(0,60),(0,60)..z0);
xy = buildcycle(z0..(60,0),(60,0)..(30,-30),(30,-30)..(-30,-30),(-30,-30)..z0);
fill xz withcolor .9white;
fill yz withcolor .94white;
fill xy withcolor .86white;
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(67,7));
label(btex $z$ etex,(7,67));
label(btex $0$ etex,(1,-6));
defaultscale := 0.8;
label(btex $yz$-plane etex,(30,30));
label(btex $xy$-plane etex,(20,-19));
label(btex $xz$-plane etex,(-30,7));
\end{mpdisplay}

\begin{mpdisplay}
a=70pt;
z0=(-a,0); z1=(a,0);
draw z0..z1 withcolor .4white;
draw (-30,-2)..(-30,2);
draw (0,-2)..(0,2);
draw (50,-2)..(50,2);
draw (-20,-2)..(-20,2);
draw (-58,-2)..(-58,2);
defaultfont := "pncr8r";
label(btex $0$ etex,(-30,-7));
label(btex $x$ etex,(70,-7));
pickup pencircle scaled 1.5pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (-20,0)..(-58,0);
drawarrow (0,0)..(50,0);
defaultfont := "ptmr8r";
label(btex $P$ etex,(0,-7));
label(btex $Q$ etex,(50,-7));
label(btex $R$ etex,(-20,-7));
label(btex $S$ etex,(-58,-7));
label.top(btex $\overrightarrow{PQ}$ etex, (25,0));
label.top(btex $\overrightarrow{RS}$ etex, (-39,0));
\end{mpdisplay}


\begin{mpdisplay}
a=70pt; b=30pt;
z0=(-b,0); z1=(a,0); z2=(0,a); z3=(0,-b);
drawarrow z0..z1 withcolor .4white;
drawarrow z3..z2 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(67,7));
label(btex $y$ etex,(7,67));
label(btex $0$ etex,(4,-6));
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (10,20)..(60,60);
drawarrow (-5,-15)..(-20,45);
drawarrow (35,15)..(55,-20);
pickup pencircle scaled 0.5pt;
dotlabel.bot(btex $P$ etex,(10,20));
label(btex $Q$ etex,(60,52));
dotlabel.rt(btex $R$ etex,(35,15));
dotlabel.bot("",(-5,-15));
label(btex $S$ etex,(55,-25));
label.top(btex $\overrightarrow{PQ}$ etex rotated 38.66, (35,40));
label.lft(btex $\overrightarrow{RS}$ etex, (47,-10));
defaultfont := "pncb8r";
label("v", (-19,15));
\end{mpdisplay}

\begin{mpdisplay}
a=70pt; b=70pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(67,7));
label(btex $z$ etex,(7,67));
label(btex $0$ etex,(1,-6));
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (10,-20)..(40,52);
drawarrow (15,60)..(-25,25);
drawarrow (0,0)..(30,-30);
pickup pencircle scaled 0.5pt;
dotlabel.bot(btex $P$ etex,(10,-20));
dotlabel.urt("",(0,0));
label(btex $Q$ etex,(46,52));
dotlabel.rt(btex $R$ etex,(15,60));
label(btex $S$ etex,(-25,20));
label.bot(btex $\overrightarrow{PQ}$ etex rotated 64.723, (33,25));
label.top(btex $\overrightarrow{RS}$ etex rotated 38.66, (-10,40));
defaultfont := "pncb8r";
label("v", (28,-20));
\end{mpdisplay}


\begin{mpdisplay}
a=5pt; b=103pt;
z0=(-a,0); z1=(b,0); z2=(0,-a); z3=(0,b);
defaultfont := "pncr8r";
for i=20 step 20 until 100: draw (0,i)..(100,i) withcolor .8white; endfor
for i=20 step 20 until 100: draw (i,0)..(i,100) withcolor .8white; endfor
for i=1 upto 4: label(decimal i,(-5,i*20)); endfor
for i=1 upto 4: label(decimal i,(i*20,-5)); endfor
drawarrow z0..z1 withcolor .4white;
drawarrow z2..z3 withcolor .4white;
label(btex $x$ etex,(105,7));
label(btex $y$ etex,(7,105));
label(btex $0$ etex,(-5,-5));
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (20,60)..(60,80);
drawarrow (60,60)..(20,40);
drawarrow (0,0)..(40,20);
defaultfont := "pncb8r";
label("u", (65,85));
label("v", (15,35));
label("w", (45,25));
pickup pencircle scaled 2.5pt;
drawdot (20,60);
drawdot (60,60);
drawdot (0,0);
\end{mpdisplay}


\begin{mpdisplay}
a=70pt; b=70pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(67,7));
label(btex $z$ etex,(7,67));
label(btex $0$ etex,(1,-6));
label(btex $P(3,4,5)$ etex,(48,46));
pickup pencircle scaled 3.5pt;
drawdot (30,40);
\end{mpdisplay}


\begin{mpdisplay}
a=70pt; b=70pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(67,7));
label(btex $z$ etex,(7,67));
label(btex $0$ etex,(1,-6));
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (0,0)..(30,40);
label.rt(btex $\textbf{v} = (3,4,5)$ etex, (31,46));
defaultfont := "pncb8r";
pickup pencircle scaled 2.5pt;
drawdot (0,0);
\end{mpdisplay}


\begin{mpdisplay}
a=5pt; b=143pt; c=50pt;
z0=(-a,0); z1=(b,0); z2=(0,-a); z3=(0,b); z4=(a,a); z5=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z2..z3 withcolor .4white;
drawarrow z4..z5 withcolor .4white;
draw (0,0)..(20,100) dashed evenly;
draw (80,30)..(100,130) dashed evenly;
draw (-60,-40)..(0,0) dashed evenly;
draw (20,-10)..(80,30) dashed evenly;
defaultfont := "pncr8r";
label(btex $y$ etex,(140,7));
label(btex $z$ etex,(7,140));
label(btex $x$ etex,(-40,-48));
label(btex $0$ etex,(-5,5));
label.top(btex $\overrightarrow{PQ}$ etex rotated 20.56, (60,115));
label.top(btex $\overrightarrow{RS}$ etex rotated 20.56, (-15,-40));
label(btex $\text{Translate~} \overrightarrow{PQ} \text{~to~} \textbf{v}$ etex,(137,75));
label(btex $\text{Translate~} \overrightarrow{RS} \text{~to~} \textbf{w}$ etex,(-60,-4.5));
pickup pencircle scaled 1.3pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (20,100)..(100,130);
drawarrow (-60,-40)..(20,-10);
drawarrow (0,0)..(80,30);
pickup pencircle scaled 2.5pt;
drawdot (0,0);
dotlabel.top(btex $P$ etex,(20,100));
label.lrt(btex $(2,1,5)$ etex,(20,100));
label.top(btex $Q$ etex,(100,130));
label.lrt(btex $(3,5,7)$ etex,(100,130));
dotlabel.top(btex $R$ etex,(-60,-40));
label.llft(btex $(1,-3,-2)$ etex,(-60,-40));
label.rt(btex $S$ etex,(23,-8));
label.lrt(btex $(2,1,0)$ etex,(20,-10));
label.bot(btex $(1,4,2)$ etex,(95,30));
defaultfont := "pncb8r";
label(btex $\textbf{v} = \textbf{w}$ etex,(97,33));
\end{mpdisplay}

\begin{mpdisplay}
a=70pt; b=70pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(67,7));
label(btex $z$ etex,(7,67));
label(btex $0$ etex,(1,-6));
defaultfont := "ptmr8r";
draw (30,55)..(30,-20);
draw (30,-20)..(50,0) dashed evenly;
draw (0,0)..(30,-20);
draw (30,-20)..(-20,-20);
label(btex $a$ etex,(-15,-9));
label(btex $Q(a,b,c)$ etex,(48,61));
label(btex $S$ etex,(-25,-19));
dotlabel.lft(btex $P$ etex,(0,0));
label(btex $R$ etex,(35,-25));
label(btex $b$ etex,(4,-26));
label(btex $c$ etex,(35,27));
pickup pencircle scaled 0.1pt;
draw (-13,-20)..(-8,-15); draw (-8,-15)..(-15,-15);
draw (43,0)..(37.5,-5); draw (37.5,-5)..(45,-5);
draw (30,-13)..(24,-9); draw (24,-9)..(24,-16);
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow z0..(30,55);
defaultfont := "pncb8r";
label("v",(10,34));
\end{mpdisplay}


\begin{mpdisplay}
a=100pt; b=85pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(97,7));
label(btex $z$ etex,(7,82));
label(btex $0$ etex,(-6,1));
defaultfont := "ptmr8r";
draw (10,30)..(10,60);
draw (10,60)..(70,60);
draw (70,60)..(70,30);
draw (70,30)..(10,30);
draw (10,60)..(30,80);
draw (30,80)..(90,80);
draw (90,80)..(90,50);
draw (90,50)..(70,30);
draw (70,60)..(90,80);
draw (10,30)..(30,50) dashed evenly withcolor .6white;
draw (30,50)..(90,50) dashed evenly withcolor .6white;
draw (30,80)..(30,-10) dashed evenly withcolor .6white;
draw (10,30)..(10,-30) dashed evenly withcolor .6white;
draw (70,30)..(70,-30) dashed evenly withcolor .6white;
draw (90,50)..(90,-10) dashed evenly withcolor .6white;
draw (30,50)..(70,30) dashed evenly withcolor .6white;
draw (30,-10)..(70,-30) dashed evenly withcolor .6white;
draw (10,-30)..(30,-10);
draw (30,-10)..(90,-10);
draw (90,-10)..(70,-30);
draw (70,-30)..(10,-30);
dotlabel.lft(btex $P(x_{\scriptscriptstyle 1},y_{\scriptscriptstyle 1},z_{\scriptscriptstyle 1})$ etex,(29,50));
dotlabel.rt(btex $Q(x_{\scriptscriptstyle 2},y_{\scriptscriptstyle 2},z_{\scriptscriptstyle 2})$ etex,(71,60));
dotlabel.rt(btex $R(x_{\scriptscriptstyle 2},y_{\scriptscriptstyle 2},z_{\scriptscriptstyle 1})$ etex,(70,30));
dotlabel.lft(btex $S(x_{\scriptscriptstyle 1},y_{\scriptscriptstyle 1},0)$ etex,(30,-10));
dotlabel.rt(btex $T(x_{\scriptscriptstyle 2},y_{\scriptscriptstyle 2},0)$ etex,(70,-30));
dotlabel.bot(btex $U(x_{\scriptscriptstyle 2},y_{\scriptscriptstyle 1},0)$ etex,(10,-30));
pickup pencircle scaled 0.1pt;
draw (17,-30)..(22,-25); draw (22,-25)..(15,-25);
draw (70,37)..(64,41); draw (64,41)..(64,34);
pickup pencircle scaled 1.2pt;
draw (30,50)..(70,60);
\end{mpdisplay}


\begin{mpdisplay}
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (0,0)..(10,30);
drawarrow (50,0)..(70,60);
drawarrow (100,0)..(130,90);
drawarrow (150,0)..(155,15);
drawarrow (210,30)..(200,0);
drawarrow (270,60)..(250,0);
pickup pencircle scaled 2.5pt;
drawdot (0,0);
drawdot (50,0);
drawdot (100,0);
drawdot (150,0);
drawdot (210,30);
drawdot (270,60);
defaultfont := "pncb8r";
label.bot("v",(0,0));
label.bot(btex $2\textbf{v}$ etex,(50,0));
label.bot(btex $3\textbf{v}$ etex,(100,0));
label.bot(btex $0.5\textbf{v}$ etex,(150,0));
label.bot(btex $-\textbf{v}$ etex,(200,0));
label.bot(btex $-2\textbf{v}$ etex,(250,0));
\end{mpdisplay}


\begin{mpdisplay}
draw (-35,0)..(85,55) withcolor white;
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (20,30)..(50,10);
drawarrow (0,10)..(30,50);
pickup pencircle scaled 2.5pt;
drawdot (20,30);
drawdot (0,10);
defaultfont := "pncb8r";
label("v", (35,12));
label("w", (15,38));
\end{mpdisplay}

\begin{mpdisplay}
draw (-15,0)..(115,55) withcolor white;
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (20,30)..(50,10);
drawarrow (50,10)..(80,50);
pickup pencircle scaled 2.5pt;
drawdot (20,30);
defaultfont := "pncb8r";
label("v", (35,12));
label("w", (67,22));
\end{mpdisplay}

\begin{mpdisplay}
draw (0,0)..(110,55) withcolor white;
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (20,30)..(50,10);
drawarrow (50,10)..(80,50);
drawarrow (20,30)..(80,50);
pickup pencircle scaled 2.5pt;
drawdot (20,30);
defaultfont := "pncb8r";
label("v", (35,12));
label("w", (67,22));
label(btex $\textbf{v} + \textbf{w}$ etex, (40,47));
\end{mpdisplay}



\begin{mpdisplay}
draw (-25,0)..(75,55) withcolor white;
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (20,30)..(50,10);
drawarrow (0,10)..(30,50);
pickup pencircle scaled 2.5pt;
drawdot (20,30);
drawdot (0,10);
defaultfont := "pncb8r";
label("v", (35,12));
label("w", (15,38));
\end{mpdisplay}


\begin{mpdisplay}
z0=(-130,-100); z1=(200,-100); z2=(235,78); z3=(-95,78);
path p;
p = buildcycle(z0..z1,z1..z2,z2..z3,z3..z0);
fill p withcolor .86white;
draw z0..z1; draw z1..z2; draw z2..z3; draw z3..z0;
pickup pencircle scaled 0.5pt;
draw (55,120)..(55,65) dashed evenly;
draw (55,120)..(-17,102) dashed evenly;
draw (53,32)..(-17,102) dashed evenly;
draw (53,32)..(53,20) dashed evenly;
draw (-17,102)..(-17,35) dashed evenly;
draw (55,65)..(-17,35) dashed evenly;
draw (53,20)..(-17,35) dashed evenly;
draw (90,-5)..(125,-60) dashed evenly;
draw (160,-5)..(125,-60) dashed evenly;
draw (55,72)..(62,70.5);
draw (62,70.5)..(62,63.5);
draw (53,27)..(60,30);
draw (60,30)..(60,23);
draw (-17,42)..(-10,40.5);
draw (-10,40.5)..(-10,33.5);
draw (125,63)..(119,65)..(115,60);
drawarrow (90,57.5)..(90,32)..(107.5,22.5);
drawarrow (101,40)..(116,30)..(132,39);
drawarrow (-17,35)..(0,-40)..(125,-60);
pickup pencircle scaled 1.2pt;
ahangle := 40;
ahlength := 6pt;
drawarrow (125,50)..(125,110);
drawarrow (125,50)..(86,89);
drawarrow (86,89)..(55,120) dashed evenly;
drawarrow (125,50)..(55,65) dashed evenly;
drawarrow (125,50)..(90,-5);
drawarrow (125,50)..(81,39);
drawarrow (81,39)..(53,32) dashed evenly;
drawarrow (125,50)..(53,20) dashed evenly;
drawarrow (125,50)..(-17,35) dashed evenly;
drawarrow (125,50)..(42,78);
drawarrow (125,50)..(160,-5);
drawarrow (125,50)..(125,-60);
drawarrow (42,78)..(-17,102) dashed evenly;
drawdot(125,50);
label.rt(btex $\textbf{u}$ etex,(125,110));
label.rt(btex $\textbf{v}$ etex,(89,89));
label.bot(btex $\textbf{w}$ etex,(81,39));
label.llft(btex $\textbf{v} + \textbf{w}$ etex,(42,78));
label.ulft(btex $\lVert\mspace{1mu}\textbf{u}\mspace{1mu}\rVert \,(\textbf{v} + \textbf{w})$ etex,(-17,102));
label.bot(btex $proj_{P} \lVert\mspace{1mu}\textbf{u}\mspace{1mu}\rVert \, \textbf{v}$ etex,(50,62));
label.bot(btex $proj_{P} \lVert\mspace{1mu}\textbf{u}\mspace{1mu}\rVert \, \textbf{w}$ etex,(53,20));
label.lft(btex $proj_{P} \lVert\mspace{1mu}\textbf{u}\mspace{1mu}\rVert \, (\textbf{v} + \textbf{w})$ etex,(-17,35));
label.ulft(btex $\lVert\mspace{1mu}\textbf{u}\mspace{1mu}\rVert \, \textbf{v}$ etex,(55,120));
label.lft(btex $\lVert\mspace{1mu}\textbf{u}\mspace{1mu}\rVert \, \textbf{w}$ etex,(53,32));
label.llft(btex $\textbf{n}(\textbf{u},\textbf{v})$ etex,(90,0));
label.lrt(btex $\textbf{n}(\textbf{u},\textbf{w})$ etex,(163,0));
label.lrt(btex $\textbf{n}(\textbf{u},\textbf{v} + \textbf{w})$ etex,(125,-60));
label(btex $\theta$ etex,(121,60));
label(btex $\theta$ etex,(61,105));
label.rt(btex $P$ etex,(218,-11));

\end{mpdisplay}


\begin{mpdisplay}
z0=(-20,-10); z1=(150,-10); z2=(185,78); z3=(15,78);
path p;
p = buildcycle(z0..z1,z1..z2,z2..z3,z3..z0);
fill p withcolor .86white;
draw z0..z1; draw z1..z2; draw z2..z3; draw z3..z0;
pickup pencircle scaled 0.5pt;
draw (55,120)..(55,65) dashed evenly;
draw (55,72)..(62,70.5);
draw (62,70.5)..(62,63.5);
draw (118,51.5)..(115,47);
draw (115,47)..(122,45.5);
draw (125,63)..(119,65)..(115,60);
drawarrow (90,57.5)..(90,32)..(107.5,22.5);
pickup pencircle scaled 1.2pt;
ahangle := 40;
ahlength := 6pt;
drawarrow (125,50)..(125,110);
drawarrow (125,50)..(86,89);
drawarrow (86,89)..(55,120) dashed evenly;
drawarrow (125,50)..(55,65) dashed evenly;
drawarrow (125,50)..(90,-5);
drawdot(125,50);
label.rt(btex $\textbf{u}$ etex,(125,110));
label.rt(btex $\textbf{v}$ etex,(89,89));
label.bot(btex $proj_{P} \lVert\mspace{1mu}\textbf{u}\mspace{1mu}\rVert \, \textbf{v}$ etex,(50,62));
label.lft(btex $\lVert\mspace{1mu}\textbf{u}\mspace{1mu}\rVert \, \textbf{v}$ etex,(55,120));
label.lft(btex $\textbf{n}(\textbf{u},\textbf{v})$ etex,(87,0));
label(btex $\theta$ etex,(121,60));
label(btex $\theta$ etex,(61,105));
label.rt(btex $P$ etex,(168,34));
\end{mpdisplay}

\begin{mpdisplay}
a=70pt; b=50pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
draw z0..(40,-25) dashed evenly withcolor .6white;
draw (40,-18)..(34,-14) withcolor .6white;
draw (34,-14)..(34,-21) withcolor .6white;
draw (-18,-25)..(-13,-20) withcolor .6white;
draw (-13,-20)..(-20,-20) withcolor .6white;
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(67,7));
label(btex $z$ etex,(7,47));
label(btex $0$ etex,(-4,3));
draw z0..(40,40);
draw (-25,-25)..(40,-25) dashed evenly withcolor .6white;
draw (40,-25)..(40,40) dashed evenly withcolor .6white;
drawarrow (-10,-10)..(0,-13)..(12,-7.5) withcolor .6white;
drawarrow (0,19)..(11,21)..(15,15) withcolor .6white;
pickup pencircle scaled 1.4pt;
drawarrow (40,40)..(56,56);
drawarrow (40,40)..(58,46);
drawarrow (40,40)..(50,24);
pickup pencircle scaled 2.5pt;
drawdot (40,40);
label.llft(btex $(x,y,z)$ etex,(35,42));
dotlabel.lrt(btex $(x,y,0)$ etex,(40,-25));
label(btex $\theta$ etex,(-1,-8));
label(btex $x$ etex,(-19,-9));
label(btex $y$ etex,(8,-31));
label.rt(btex $z$ etex,(40,10));
label.bot(btex $\rho$ etex,(25,20));
label.bot(btex $\phi$ etex,(6,20));
label.rt(btex $\textbf{e}_{\rho}$ etex,(56,56));
label.rt(btex $\textbf{e}_{\theta}$ etex,(58,46));
label.rt(btex $\textbf{e}_{\phi}$ etex,(50,24));
\end{mpdisplay}


\begin{mpdisplay}
a=70pt; b=50pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
draw z0..(40,-25) dashed evenly withcolor .6white;
draw (40,-18)..(34,-14) withcolor .6white;
draw (34,-14)..(34,-21) withcolor .6white;
draw (-18,-25)..(-13,-20) withcolor .6white;
draw (-13,-20)..(-20,-20) withcolor .6white;
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(67,7));
label(btex $z$ etex,(7,47));
label(btex $0$ etex,(-4,3));
draw (-25,-25)..(40,-25) dashed evenly withcolor .6white;
draw (40,-25)..(40,40);
drawarrow (-10,-10)..(0,-13)..(12,-7.5) withcolor .6white;
pickup pencircle scaled 1.4pt;
drawarrow (40,40)..(56,30);
drawarrow (40,40)..(58,46);
drawarrow (40,40)..(40,60);
pickup pencircle scaled 2.5pt;
dotlabel.llft(btex $(x,y,z)$ etex,(40,40));
dotlabel.lrt(btex $(x,y,0)$ etex,(40,-25));
label(btex $\theta$ etex,(-1,-8));
label(btex $x$ etex,(-19,-9));
label(btex $y$ etex,(8,-31));
label.lft(btex $z$ etex,(40,15));
label.top(btex $r$ etex,(20,-12));
label.rt(btex $\textbf{e}_{r}$ etex,(56,30));
label.rt(btex $\textbf{e}_{\theta}$ etex,(58,46));
label.top(btex $\textbf{e}_{z}$ etex,(40,60));
\end{mpdisplay}


\begin{mpdisplay}
z0=(-30,0); z1=(70,0); z2=(0,50); z3=(0,-6);
drawarrow z0..z1 withcolor .4white;
drawarrow z3..z2 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(67,7));
label(btex $y$ etex,(7,47));
label(btex $0$ etex,(4,-6));
drawarrow (60,20)..(22,20);
drawarrow (-20,20)..(18,20);
drawarrow (20,40)..(20,22);
drawarrow (20,-6)..(20,18);
drawarrow (40,40)..(22,22);
drawarrow (-10,40)..(5,25)..(18,18);
pickup pencircle scaled 0.5pt;
dotlabel.lrt(btex $(a,b)$ etex,(20,20));
\end{mpdisplay}

\begin{mpdisplay}
a=120pt; b=70pt; c=30pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-19,-27));
label(btex $y$ etex,(117,7));
label(btex $z$ etex,(7,67));
label(btex $0$ etex,(1,-6));
draw (120,60)..(-30,30);
label.lrt(btex $L$ etex,(-30,30));
pickup pencircle scaled 1.2pt;
ahangle := 40;
ahlength := 6pt;
drawarrow z0..(19,38);
drawarrow (20,40)..(69,49.8);
drawarrow z0..(69,49.8);
drawarrow z0..(110,58) dashed evenly;
pickup pencircle scaled 2.5pt;
drawdot (0,0);
dotlabel.ulft(btex $P_{\scriptscriptstyle 1}(x_{\scriptscriptstyle 1},y_{\scriptscriptstyle 1},z_{\scriptscriptstyle 1})$ etex,(20,40));
dotlabel.top(btex $P_{\scriptscriptstyle 2}(x_{\scriptscriptstyle 2},y_{\scriptscriptstyle 2},z_{\scriptscriptstyle 2})$ etex,(70,50));
defaultfont := "pncb8r";
label(btex $\textbf{r}_{\scriptscriptstyle 1}$ etex,(6,25));
label(btex $\textbf{r}_{\scriptscriptstyle 2}$ etex,(31,30));
label(btex $\textbf{r}_{\scriptscriptstyle 2} - \textbf{r}_{\scriptscriptstyle 1}$ etex,(37,50));
label.rt(btex $\textbf{r}_{\scriptscriptstyle 1} + t(\textbf{r}_{\scriptscriptstyle 2} - \textbf{r}_{\scriptscriptstyle 1})$ etex,(57,25));
\end{mpdisplay}

\begin{mpdisplay}
z0=(0,0); z1=(100,0); z2=(135,35); z3=(35,35);
path p;
p = buildcycle(z0..z1,z1..z2,z2..z3,z3..z0);
fill p withcolor .86white;
draw z0..z1; draw z1..z2; draw z2..z3; draw z3..z0;
pickup pencircle scaled 0.5pt;
draw (90,25)..(90,55) dashed evenly;
draw (90,25)..(50,15) dashed evenly;
draw (50,45)..(90,55) dashed evenly;
draw (90,32)..(84,30.5);
draw (84,30.5)..(84,23.5);
draw (50,38)..(56,39.5);
draw (56,39.5)..(56,46.5);
draw (50,28)..(56,30)..(60,25);
draw (42,30){dir -45}..{dir -90}(44,25){dir -90}..{dir -90}(44,20){dir -90}..{dir -5}(46,15);
draw (42,30){dir 45}..{dir 90}(44,35){dir 90}..{dir 90}(44,40){dir 90}..{dir 5}(46,44);
pickup pencircle scaled 1.2pt;
ahangle := 40;
ahlength := 6pt;
drawarrow (50,15)..(50,60);
drawarrow (50,15)..(89,54);
dotlabel.urt(btex $Q$ etex, (90,55));
dotlabel.bot(btex $R$ etex, (50,15));
label.rt(btex $\textbf{n}$ etex,(50,60));
label.rt(btex $\textbf{r}$ etex,(63,40));
label.rt(btex $D$ etex,(90,40));
label(btex $\theta$ etex,(54,25));
label.lft(btex $D$ etex,(43,30));
label.rt(btex $P$ etex,(118,18));
\end{mpdisplay}


\begin{mpdisplay}
a=70pt; b=50pt; c=38pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
draw z0..(40,-25) dashed evenly withcolor .6white;
draw (40,-18)..(34,-14) withcolor .6white;
draw (34,-14)..(34,-21) withcolor .6white;
draw (-18,-25)..(-13,-20) withcolor .6white;
draw (-13,-20)..(-20,-20) withcolor .6white;
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-27,-35));
label(btex $y$ etex,(67,7));
label(btex $z$ etex,(7,47));
label(btex $0$ etex,(-4,3));
draw (-25,-25)..(40,-25) dashed evenly withcolor .6white;
draw (40,-25)..(40,40);
drawarrow (-10,-10)..(0,-13)..(12,-7.5) withcolor .6white;
pickup pencircle scaled 2.5pt;
dotlabel.urt(btex $P(x,y,z)$ etex,(40,40));
dotlabel.lrt(btex $P_{\scriptscriptstyle 0}(x,y,0)$ etex,(40,-25));
label(btex $\theta$ etex,(-1,-8));
label(btex $x$ etex,(-19,-9));
label(btex $y$ etex,(8,-31));
label.rt(btex $z$ etex,(40,20));
label.top(btex $r$ etex,(20,-12));
\end{mpdisplay}

\begin{mpdisplay}
a=100pt; b=50pt; c=50pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
path P;
P = buildcycle((-30,-10)..(-15,25),(-15,25)..(55,-5),(55,-5)..(40,-40),(40,-40)..(-30,-10));
fill P withcolor .86white;
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
draw (3,7)..(10,4) withcolor .6white;
draw (10,4)..(7,-3) withcolor .6white;
draw (7,-3)..z0 withcolor .6white;
draw (7,-3)..(35,-15) dashed evenly withcolor .6white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-39,-47));
label(btex $y$ etex,(97,7));
label(btex $z$ etex,(7,47));
label(btex $0$ etex,(1,-6));
ahangle := 40;
drawarrow (-9,15)..(-16.4,6.1)..(-16.6,-5.47) withcolor .6white;
label(btex $\theta$ etex,(-10,5));
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
drawarrow z0..(-15,25);
drawarrow z0..(-30,-10);
drawarrow z0..(18,42);
drawarrow z0..(-18,-42) dashed evenly;
pickup pencircle scaled 2.5pt;
drawdot (0,0);
defaultfont := "pncb8r";
label("v",(-20,30));
label("w",(-35,-15));
label(btex $\textbf{v} \pmb{\times} \textbf{w}$ etex,(33,42));
label(btex $-\textbf{v} \pmb{\times} \textbf{w}$ etex,(-8,-47));
label(btex $P$ etex rotated -23,(34,-28));
\end{mpdisplay}


\begin{mpdisplay}
ahangle := 40;
pickup pencircle scaled 0.5pt;
drawdblarrow (25,0)..(22.36,11.18)..(15,20)..(0,25)..(-15,20) withcolor .6white;
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
drawarrow (0,0)..(50,0);
drawarrow (0,0)..(-30,40);
pickup pencircle scaled 2.5pt;
drawdot (0,0);
label(btex $90^{\circ} < \theta \le 180^{\circ}$ etex,(40,30));
defaultfont := "pncb8r";
label("v",(56,0));
label("w",(-38,40));
\end{mpdisplay}


\begin{mpdisplay}
a=150pt; b=70pt; c=30pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
ahangle := 40;
drawdblarrow (24.25,6.06)..(19.78,15.29)..(11.97,21.94) withcolor .6white;
label(btex $\theta$ etex,(24,19));
defaultfont := "pncr8r";
label(btex $x$ etex,(-19,-27));
label(btex $y$ etex,(147,7));
label(btex $z$ etex,(7,67));
label(btex $0$ etex,(1,-6));
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
drawarrow z0..(30,55);
drawarrow z0..(120,30);
drawarrow (120,30)..(30.5,55.5);
pickup pencircle scaled 2.5pt;
drawdot (0,0);
defaultfont := "pncb8r";
label("v",(14,36));
label("w",(60,9));
label(btex $\textbf{v} - \textbf{w}$ etex,(85,49));
\end{mpdisplay}


\begin{mpdisplay}
a=150pt; b=70pt; c=28pt;
z0=(0,0); z1=(a,0); z2=(0,b); z3=(-c,-c);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
drawarrow z0..z3 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(-17,-25));
label(btex $y$ etex,(147,7));
label(btex $z$ etex,(7,67));
label(btex $0$ etex,(1,-6));
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow z0..(30,55);
drawarrow z0..(120,30);
drawarrow (31.5,56.5)..(121.5,31.5);
pickup pencircle scaled 2.5pt;
drawdot (0,0);
dotlabel.urt(btex $P(x_{\scriptstyle 1},y_{\scriptstyle 1},z_{\scriptstyle 1})$ etex,(31.5,56.5));
dotlabel.rt(btex $Q(x_{\scriptstyle 2},y_{\scriptstyle 2},z_{\scriptstyle 2})$ etex,(121.5,31.5));
defaultfont := "pncb8r";
label("v",(10,28));
label("w",(60,9));
label(btex $\textbf{w} - \textbf{v}$ etex,(96,46));
\end{mpdisplay}

\begin{mpdisplay}
a=150pt; b=70pt;
z0=(0,0); z1=(a,0); z2=(0,b);
drawarrow z0..z1 withcolor .4white;
drawarrow z0..z2 withcolor .4white;
defaultfont := "pncr8r";
label(btex $x$ etex,(147,7));
label(btex $y$ etex,(7,67));
label(btex $0$ etex,(-6,-6));
defaultfont := "ptmr8r";
draw (40,0)..(40,40) dashed evenly withcolor .6white;
draw (0,40)..(40,40) dashed evenly withcolor .6white;
draw (80,0)..(80,20) dashed evenly withcolor .6white;
draw (0,20)..(80,20) dashed evenly withcolor .6white;
draw (120,0)..(120,20) dashed evenly withcolor .6white;
draw (0,60)..(120,60) dashed evenly withcolor .6white;
draw (120,20)..(120,60) withcolor .6white;
draw (80,20)..(120,20) withcolor .6white;
label.lft(btex $w_{\scriptstyle 2}$ etex, (0,40));
label.lft(btex $v_{\scriptstyle 2}$ etex, (0,20));
label.bot(btex $w_{\scriptstyle 1}$ etex, (40,0));
label.bot(btex $v_{\scriptstyle 1}$ etex, (80,0));
label(btex $v_{\scriptstyle 1} + w_{\scriptstyle 1}$ etex, (120,-5));
label.lft(btex $v_{\scriptstyle 2} + w_{\scriptstyle 2}$ etex, (0,60));
label.rt(btex $w_{\scriptstyle 2}$ etex, (120,40));
label.bot(btex $w_{\scriptstyle 1}$ etex, (100,20));
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (80,20)..(120,60) withcolor .6white;
drawarrow (40,40)..(120,60) withcolor .6white;
drawarrow z0..(80,20);
drawarrow z0..(40,40);
drawarrow (0,0)..(120,60);
pickup pencircle scaled 0.1pt;
draw (113,20)..(113,27) withcolor .6white;
draw (113,27)..(120,27) withcolor .6white;
pickup pencircle scaled 2.5pt;
drawdot (0,0);
defaultfont := "pncb8r";
label("v", (60,10));
label("v", (78,53));
label("w", (23,30));
label("w", (107,40));
label(btex $\textbf{v} + \textbf{w}$ etex, (60,30));
\end{mpdisplay}


\begin{mpdisplay}
draw (-45,-35)..(85,35) withcolor white;
pickup pencircle scaled 1.2pt;
ahlength := 6pt;
ahangle := 40;
drawarrow (20,30)..(50,10);
drawarrow (50,10)..(20,-30);
drawarrow (20,30)..(20,-30);
pickup pencircle scaled 2.5pt;
drawdot (20,30);
defaultfont := "pncb8r";
label("v", (35,12));
label(btex $-\textbf{w}$ etex, (45,-10));
label(btex $\textbf{v} - \textbf{w}$ etex, (5,0));
\end{mpdisplay}




\end{document}