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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Lines and Planes" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 13 "Description: " }
{TEXT -1 17 "Parametric line (" }{TEXT 257 9 "animation" }{TEXT -1 53 
"), examples of planes, plane and its normal vectors (" }{TEXT 258 9 "
animation" }{TEXT -1 30 "), plane trough three points (" }{TEXT 260 9 
"animation" }{TEXT -1 39 "), line of intersection of two planes (" }
{TEXT 259 9 "animation" }{TEXT -1 2 ")." }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 4 "Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plo
ttools):\nwith(plots):" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, the na
mes arrow and changecoords have been redefined\n" }}}{SECT 1 {PARA 5 "
" 0 "" {TEXT -1 15 "Parametric Line" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "rv:=arrow(<2,1,3>,color=co
ral,width=[0.1,relative=false],\nhead_width=[0.3,relative=false]):\npt
:=sphere([2,1,3],0.1,color=green,style=patchnogrid):\ndirv:=arrow(<2,1
,3>,<1,2,-1>,color=violet,width=[0.05,relative=false],head_width=[0.3,
relative=false]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "linet
:=display([seq(line([2,1,3],[2+t/10,1+2*t/10,3-t/10],color=red,thickne
ss=3),t=0..20),\nseq(line([4,5,1],[2-t/10,1-2*t/10,3+t/10],color=red,t
hickness=3),t=0..20)],\ninsequence=true):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 199 "parline:=display(\{rv,pt,linet,dirv\},scaling=cons
trained,view=[-3..5,-3..5,-3..5],\naxes=normal,labels=[`x`,`y`,`z`],\n
orientation=[-10,65],title=`Line through P(2,1,3) in the direction of \+
v=<1,2,-1>`):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 
0 "" {TEXT -1 6 "Planes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "
hplanes:=display(\{plot3d(-1,x=-2..2,y=-2..2,color=red,grid=[9,9]),\np
lot3d(0,x=-2..2,y=-2..2,color=violet,grid=[9,9]),\nplot3d(1,x=-2..2,y=
-2..2,color=coral,grid=[9,9])\},title=`Horizontal planes z=-1, z=0, an
d z=1`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "firstoctant:=d
isplay(\{\nplot3d([0,y,z],y=0..4,z=0..4,color=violet,grid=[5,5],style=
wireframe),\nplot3d([x,0,z],x=0..4,z=0..4,color=coral,grid=[5,5],style
=wireframe),\nplot3d([x,y,0], y=0..4, x=0..4,color=red,grid=[5,5],styl
e=wireframe)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plane1
:=plot3d(1 - x - y,x=-2..2,y=-2..2,color=blue,grid=[5,5],\ntitle=`The \+
plane z=1-x-y`,axes=normal):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 383 "plane1a[1]:=display(\{firstoctant,plane1,\nline([1,0,0],[0,0,1]
,color=magenta,thickness=2),\nline([1,0,0],[0,1,0],color=magenta,thick
ness=2),\nline([0,1,0],[0,0,1],color=magenta,thickness=2)\n\},view=[0.
.2,0..2,0..2],title=`The plane z=1-x-y in the first octant`):\nplane1a
[2]:=display(\{plane1a[1]\},view=[0..2,0..2,0..2],\ntitle=`The plane z
=1-x-y in the first octant`,orientation=[-35,60]):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 480 "plane1a[3]:=display(\{plane1a[2],\narrow(<1,0,0>,<
1,1,1>,color=red,width=[0.05,relative=false],head_width=[0.2,relative=
false]),\narrow(<0,1,0>,<1,1,1>,color=red,width=[0.05,relative=false],
head_width=[0.2,relative=false]),\narrow(<0,0,1>,<1,1,1>,color=red,wid
th=[0.05,relative=false],head_width=[0.2,relative=false]),\narrow(<1/4
,1/4,1/2>,<1,1,1>,color=red,width=[0.05,relative=false],head_width=[0.
2,relative=false])\n\},orientation=[-35,60],title=`The plane z=1-x-y a
nd its normals`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 446 "plane
2:=plot3d(-(x+2*y-6)/3,x=-2..2,y=-2..2,color=coral,style=patchnogrid):
\nplane2a[1]:=display(sphere([0,0,2],0.1,color=blue,style=patchnogrid)
,\ntitle=`Plane can be defined by a point ...`):\nplane2a[2]:=display(
\{plane2a[1],\narrow(<0,0,2>,<1,2,3>,color=red,width=[0.1,relative=fal
se],head_width=[0.3,relative=false])\},title=`... and a normal vector`
):\nplane2a[3]:=display(\{plane2a[2],plane2\},title=`Plane through (0,
0,2) with normal n=<1,2,3>`):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 577 "point3a:=sphere([1,3,2],0.2,color=green,style=patchnogrid):\n
point3b:=sphere([3,-1,6],0.2,color=green,style=patchnogrid):\npoint3c:
=sphere([5,2,0],0.2,color=green,style=patchnogrid):\nvab:=arrow(<1,3,2
>,<2,-4,4>,color=coral,\nwidth=[0.08,relative=false],head_width=[0.2,r
elative=false]):\nvac:=arrow(<1,3,2>,<4,-1,-2>,color=coral,\nwidth=[0.
08,relative=false],head_width=[0.2,relative=false]):\nn:=arrow(<1,3,2>
,<12/4,20/4,14/4>,color=red,\nwidth=[0.08,relative=false],head_width=[
0.2,relative=false]):\nplane3:=plot3d(-6/7*x-10/7*y+50/7,x=-2..6,y=-2.
.6,color=violet,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 454 "plane3a[1]:=display(\{point3a,point3b,point3c\},\nti
tle=`Plane through P(1,3,2), Q(3,-1,6), and R(5,2,0)`):\nplane3a[2]:=d
isplay(\{point3a,point3b,point3c,vab,vac\},\ntitle=`Plane through P(1,
3,2), Q(3,-1,6), and R(5,2,0)`):\nplane3a[3]:=display(\{point3a,point3
b,point3c,vab,vac,n\},\ntitle=`Plane through P(1,3,2), Q(3,-1,6), and \+
R(5,2,0)`):\nplane3a[4]:=display(\{point3a,point3b,point3c,vab,vac,n,p
lane3\},\ntitle=`Plane through P(1,3,2), Q(3,-1,6), and R(5,2,0)`):" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 
30 "Line of Intersection of Planes" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 566 "plane4a:=plot3d(-x+2*y-1,x=-1..3,y=-1..3,color=green
,grid=[5,5]):\nplane4b:=plot3d(x/2-y+1/2,x=-1..3,y=-1..3,color=violet,
grid=[5,5]):\nline4:=spacecurve([1+2*t,1+t,0],t=-1..1,thickness=3,colo
r=red):\npoint4:=sphere([1,1,0],0.2,color=coral,style=patchnogrid):\nn
4a:=arrow(<1,1,0>,<1,-2,1>,color=green,\nwidth=[0.08,relative=false],h
ead_width=[0.2,relative=false]):\nn4b:=arrow(<1,1,0>,<1,-2,-2>,color=v
iolet,\nwidth=[0.08,relative=false],head_width=[0.2,relative=false]):
\nv:=arrow(<1,1,0>,<2,1,0>,color=magenta,\nwidth=[0.08,relative=false]
,head_width=[0.2,relative=false]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 121 "plane4[1]:=display(\{plane4a,plane4b,line4,point4\},
scaling=constrained,\ntitle=`Intersection of x-2y+z=-1 and x-2y-2z=-1`
):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "plane4[2]:=display(\{plane4a
,plane4b,point4\},\nscaling=constrained,style=wireframe,title=`Point P
(1,1,0) is common for both planes`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 136 "plane4[3]:=display(\{plane4a,plane4b,point4,n4a\},\nscaling=con
strained,style=wireframe,title=`Vector n1=<1,-2,1> is normal to x-2y+z
=-1`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "plane4[4]:=display(\{pla
ne4a,plane4b,point4,n4a,n4b\},\nscaling=constrained,style=wireframe,ti
tle=`Vector n2=<1,-2,-2> is normal to x-2y-2z=-1`):" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 133 "plane4[5]:=display(\{plane4a,plane4b,point4,n4a,n
4b,v\},\nscaling=constrained,style=wireframe,title=`Their cross-produc
t is v=<2,1,0>`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "plane4[6]:=di
splay(\{plane4a,plane4b,point4,n4a,n4b,v,line4\},\nscaling=constrained
,style=wireframe,title=`Intersection of x-2y+z=-1 and x-2y-2z=-1`):" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
5 "Plots" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 15 "Parametric Line" }}
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