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0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 555 "" 0 1 0 0 0 0 0 1 0 0 0 0 
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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 58 "Vectors in two and three dimensio
ns in terms of components" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Sto
ne, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  26
.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Vectors in the coordinate p
lane " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 95 "The position vector of a point in the coordinate plane \+
can be expressed using the two standard " }{TEXT 259 13 "basis vectors
" }{TEXT -1 1 " " }{TEXT 263 1 "i" }{TEXT -1 5 " and " }{TEXT 264 1 "j
" }{TEXT -1 65 " of unit length, and which lie in the directions of th
e positive " }{TEXT 265 1 "x" }{TEXT -1 5 " and " }{TEXT 266 1 "y" }
{TEXT -1 20 " axes respectively. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus \+
" }{TEXT 267 1 "i" }{TEXT -1 5 " and " }{TEXT 268 1 "j" }{TEXT -1 39 "
 are the position vectors of the points" }{XPPEDIT 18 0 " ``(1,0)" "6#
-%!G6$\"\"\"\"\"!" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(0,1)" "6#-%!
G6$\"\"!\"\"\"" }{TEXT -1 15 " respectively. " }}{PARA 0 "" 0 "" 
{TEXT -1 23 "For example, the point " }{XPPEDIT 18 0 "P(3,2);" "6#-%\"
PG6$\"\"$\"\"#" }{TEXT -1 27 " has the position vector 3 " }{TEXT 271 
1 "i" }{TEXT -1 5 " + 2 " }{TEXT 272 1 "j" }{TEXT -1 2 ". " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 400 300 300 {PLOTDATA 2 "60-%'CU
RVESG6'7$7$$\"\"!F)F(7$$\"\"$F)$\"\"#F)7%7$$\"+#*>\"y&G!\"*$\"+7?G`>F3
F*7$$\"+3!)=-HF3$\"+))zr')=F3-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&%$RGB
G$\"#&)!\"#F)FC-%*THICKNESSG6#F.-F$6'7$F'7$$\"\"\"F)F(7%7$$\"+++++#*!#
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6&7$$\"#KF_q$\"#AF_qQ)P(~3,2~)FbqFjoF[s-%+AXESLABELSG6%%\"xG%\"yG-Fdq6
#%(DEFAULTG-%*AXESTICKSG6$F^q\"\"%-%%VIEWG6$;$!\"$F_q$F`tF);FetF+" 1 
2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve
 9" "Curve 10" "Curve 11" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 21 "In general, the point" }{XPPEDIT 18 0 " ``(x[1]
,y[1])" "6#-%!G6$&%\"xG6#\"\"\"&%\"yG6#F)" }{TEXT -1 25 " has the posi
tion vector " }{TEXT 273 1 "v" }{XPPEDIT 18 0 "``[1] = x[1];" "6#/&%!G
6#\"\"\"&%\"xG6#F'" }{TEXT -1 1 " " }{TEXT 270 1 "i" }{TEXT -1 3 " + \+
" }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 269 
1 "j" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"x[1" "6#&%\"xG6#\"\"\"" }{TEXT -1 8 " is the " }{TEXT 274 2 "x " }
{TEXT 259 11 "- component" }{TEXT -1 4 " of " }{TEXT 275 1 "v" }
{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 8 " is the " }{TEXT 276 1 "y
" }{TEXT -1 1 " " }{TEXT 259 11 "- component" }{TEXT -1 4 " of " }
{TEXT 277 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 2 ".
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The \+
vector  " }{TEXT 290 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }
{TEXT -1 28 " can now be considered as a " }{TEXT 259 11 "free vector
" }{TEXT -1 78 ", that is, its initial point can be located anywhere i
n the coordinate plane. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 25 "Given two (free) vectors " }{TEXT 280 1 "v" }
{XPPEDIT 18 0 "``[1] = x[1];" "6#/&%!G6#\"\"\"&%\"xG6#F'" }{TEXT -1 1 
" " }{TEXT 281 1 "i" }{XPPEDIT 18 0 "``+y[1];" "6#,&%!G\"\"\"&%\"yG6#F
%F%" }{TEXT -1 1 " " }{TEXT 282 1 "j" }{TEXT -1 5 " and " }{TEXT 283 
1 "v" }{XPPEDIT 18 0 "``[2] = x[2];" "6#/&%!G6#\"\"#&%\"xG6#F'" }
{TEXT -1 1 " " }{TEXT 284 1 "i" }{XPPEDIT 18 0 "``+y[2]" "6#,&%!G\"\"
\"&%\"yG6#\"\"#F%" }{TEXT -1 1 " " }{TEXT 285 1 "j" }{TEXT -1 15 ", th
eir sum is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 286 1 "v" }
{XPPEDIT 18 0 "``[1]+``" "6#,&&%!G6#\"\"\"F'F%F'" }{TEXT 287 1 "v" }
{XPPEDIT 18 0 "``[2] = ``(x[1]+x[2])" "6#/&%!G6#\"\"#-F%6#,&&%\"xG6#\"
\"\"F.&F,6#F'F." }{TEXT -1 1 " " }{TEXT 288 1 "i" }{XPPEDIT 18 0 "`` +
 ``(y[1]+y[2])" "6#,&%!G\"\"\"-F$6#,&&%\"yG6#F%F%&F*6#\"\"#F%F%" }
{TEXT -1 1 " " }{TEXT 289 1 "j" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 13 "that is, the " }{TEXT 278 1 "x" }{TEXT -1 5 " and " }
{TEXT 279 1 "y" }{TEXT -1 22 " components are added." }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{GLPLOT2D 418 322 322 {PLOTDATA 2 "6<-%'CURVESG6&
7$7$$\"\"!F)F(7$$\"\"&F)$\"\"#F)7%7$$\"+#zJ%=Z!\"*$\"+>0#*y>F3F*7$$\"+
3#o:y%F3$\"+\"[z5#=F3-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&%$RGBG$\"#&)!
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&FB$F,FZF)\"\"\"-F$6&7$F*7$FJF-7%7$$\"++++]yF3$\"++++&3#F3F\\r7$F_r$\"
++++:>F3F;Fep-F$6&7$F\\rFI7%7$$\"++++:zF3$\"+++++xF3FI7$$\"++++&3)F3F]
sF;Feq-%%TEXTG6&7$$\"$L$FE$\"$B%FEQ(v~~~+~v6\"F_o-%%FONTG6%%*HELVETICA
G%%BOLDG\"#5-Fcs6&7$$\"#NFZ$\"#TFZQ,1~~~~~~~~~2F[tF_o-F]t6$F_tFK-Fcs6&
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F_u$!#DFEQ\"iF[tFepF\\t-Fcs6&7$F^v$\"$v\"FEF_wFepF\\t-Fcs6&7$$\"#FFZF]
wQ\"xF[tFep-F]t6$F_tFat-Fcs6&7$$\"#iFZFcwFjwFepF[x-Fcs6&7$$\"$&GFE$!\"
%FZFjuFepFjt-Fcs6&7$$\"$N'FE$\"#;FZFivFepFjt-Fcs6&7$$\"$X&FEFauQ\"jF[t
FeqF\\t-Fcs6&7$$\"$X)FEF`vFeyFeqF\\t-Fcs6&7$$\"$:&FEFauQ\"yF[tFeqF[x-F
cs6&7$$\"$:)FEF`vF`zFeqF[x-Fcs6&7$$\"#`FZFCFjuFeqFjt-Fcs6&7$$\"#$)FZ$
\"$X%FEFivFeqFjt-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17
" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C
urve 24" "Curve 25" }}}{PARA 0 "" 0 "" {TEXT -1 11 "Similarly, " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 291 1 "v" }{XPPEDIT 18 0 "``[1
]-``;" "6#,&&%!G6#\"\"\"F'F%!\"\"" }{TEXT 292 1 "v" }{XPPEDIT 18 0 "``
[2] = ``(x[1]-x[2]);" "6#/&%!G6#\"\"#-F%6#,&&%\"xG6#\"\"\"F.&F,6#F'!\"
\"" }{TEXT -1 1 " " }{TEXT 293 1 "i" }{XPPEDIT 18 0 "``+``(y[1]-y[2]);
" "6#,&%!G\"\"\"-F$6#,&&%\"yG6#F%F%&F*6#\"\"#!\"\"F%" }{TEXT -1 1 " " 
}{TEXT 294 1 "j" }{TEXT -1 3 ",  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "and for a scalar " }{TEXT 315 1 "s" }
{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{TEXT 316 1 "s" }
{TEXT -1 1 " " }{TEXT 317 1 "v" }{XPPEDIT 18 0 "``[1] = s*x[1]" "6#/&%
!G6#\"\"\"*&%\"sGF'&%\"xG6#F'F'" }{TEXT -1 1 " " }{TEXT 318 1 "i" }
{XPPEDIT 18 0 " ``+ s*y[1]" "6#,&%!G\"\"\"*&%\"sGF%&%\"yG6#F%F%F%" }
{TEXT -1 1 " " }{TEXT 319 1 "j" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 6 "length
" }{TEXT -1 19 ", or magnitude, || " }{TEXT 323 1 "v" }{XPPEDIT 18 0 "
``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 16 " || of a vector " }{TEXT 320 1 "
v" }{XPPEDIT 18 0 "``[1] = x[1];" "6#/&%!G6#\"\"\"&%\"xG6#F'" }{TEXT 
-1 1 " " }{TEXT 321 1 "i" }{XPPEDIT 18 0 "``+y[1];" "6#,&%!G\"\"\"&%\"
yG6#F%F%" }{TEXT -1 1 " " }{TEXT 322 1 "j" }{TEXT -1 4 " is " }}{PARA 
256 "" 0 "" {TEXT -1 4 " || " }{TEXT 324 1 "v" }{XPPEDIT 18 0 "``[1]" 
"6#&%!G6#\"\"\"" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``= sqrt(x[1]^2+y[1
]^2)" "6#/%!G-%%sqrtG6#,&*$&%\"xG6#\"\"\"\"\"#F-*$&%\"yG6#F-F.F-" }
{TEXT -1 3 ".  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 402 16 "___
_____________" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 16 "For example, if " }{TEXT 295 1 "v" }
{XPPEDIT 18 0 "``[1]=2" "6#/&%!G6#\"\"\"\"\"#" }{TEXT -1 1 " " }{TEXT 
296 1 "i" }{XPPEDIT 18 0 " ``- 4" "6#,&%!G\"\"\"\"\"%!\"\"" }{TEXT -1 
1 " " }{TEXT 297 1 "j" }{TEXT -1 5 " and " }{TEXT 298 1 "v" }{XPPEDIT 
18 0 "``[2] = 5" "6#/&%!G6#\"\"#\"\"&" }{TEXT -1 1 " " }{TEXT 299 1 "i
" }{XPPEDIT 18 0 "`` + 3" "6#,&%!G\"\"\"\"\"$F%" }{TEXT -1 1 " " }
{TEXT 300 1 "j" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{TEXT 301 1 "v" }{XPPEDIT 18 0 "``[1]+``" "6#,&&%!G6#\"\"\"F'F%F'" }
{TEXT 302 1 "v" }{XPPEDIT 18 0 "``[2] = 7;" "6#/&%!G6#\"\"#\"\"(" }
{TEXT -1 1 " " }{TEXT 303 1 "i" }{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\"
F$!\"\"" }{TEXT -1 1 " " }{TEXT 304 1 "j" }{TEXT -1 2 ", " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{TEXT 305 1 "v" }{XPPEDIT 18 0 "``[1]-``;
" "6#,&&%!G6#\"\"\"F'F%!\"\"" }{TEXT 306 1 "v" }{XPPEDIT 18 0 "``[2] =
 -3;" "6#/&%!G6#\"\"#,$\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 307 1 "i" }
{XPPEDIT 18 0 "``-7;" "6#,&%!G\"\"\"\"\"(!\"\"" }{TEXT -1 1 " " }
{TEXT 308 1 "j" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 3 " ( " }{TEXT 
309 1 "v" }{XPPEDIT 18 0 "``[1]+``;" "6#,&&%!G6#\"\"\"F'F%F'" }{TEXT 
310 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 2 " )" }
{XPPEDIT 18 0 " ``= 7/2" "6#/%!G*&\"\"(\"\"\"\"\"#!\"\"" }{TEXT -1 1 "
 " }{TEXT 311 1 "i" }{XPPEDIT 18 0 "``-1/2;" "6#,&%!G\"\"\"*&F%F%\"\"#
!\"\"F(" }{TEXT -1 1 " " }{TEXT 312 1 "j" }{TEXT -1 2 ", " }}{PARA 
256 "" 0 "" {TEXT -1 4 " || " }{TEXT 313 1 "v" }{XPPEDIT 18 0 "``[1]+`
`;" "6#,&&%!G6#\"\"\"F'F%F'" }{TEXT 314 1 "v" }{XPPEDIT 18 0 "``[2];" 
"6#&%!G6#\"\"#" }{TEXT -1 6 " || = " }{XPPEDIT 18 0 "sqrt(49+1) = sqrt
(50)" "6#/-%%sqrtG6#,&\"#\\\"\"\"F)F)-F%6#\"#]" }{XPPEDIT 18 0 "`` = 5
sqrt(2)" "6#/%!G*&\"\"&\"\"\"-%%sqrtG6#\"\"#F'" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Other not
ations for vectors in terms of components are  " }{TEXT 325 1 "v" }
{XPPEDIT 18 0 "``[1]=2" "6#/&%!G6#\"\"\"\"\"#" }{TEXT -1 1 " " }{TEXT 
326 1 "i" }{XPPEDIT 18 0 " ``- 4" "6#,&%!G\"\"\"\"\"%!\"\"" }{TEXT -1 
1 " " }{TEXT 327 1 "j" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "[2,-4] = matr
ix([[2],[-4]])" "6#/7$\"\"#,$\"\"%!\"\"-%'matrixG6#7$7#F%7#,$F'F(" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "The last two representat
ions are called " }{TEXT 259 22 "row and column vectors" }{TEXT -1 15 
" respectively. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 93 "The above calculations can be performed in Maple using th
e \"old\" vector structure as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "v1 := vector([2,-4]);\n
v2 := vector([5,3]);\nv1 + v2 = evalm(v1 + v2);\nv1 - v2 = evalm(v1 - \+
v2);\n1/2*(v1 + v2) = evalm(1/2*(v1 + v2));\nlinalg[norm](v1+v2,2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G-%'vectorG6#7$\"\"#!\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G-%'vectorG6#7$\"\"&\"\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%#v1G\"\"\"%#v2GF&-%'vectorG6#7$\"
\"(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%#v1G\"\"\"%#v2G!\"\"-%
'vectorG6#7$!\"$!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%#v1G#\"\"
\"\"\"#*&F&F'%#v2GF'F'-%'vectorG6#7$#\"\"(F(#!\"\"F(" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"#\"\"\"\"\"&" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Alternatively, using the \+
\"new\" (column) Vector structure we have . . . " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "v1 := Vecto
r([2,-4]);\nv2 := Vector([5,3]);\nv1 + v2;\nv1 - v2;\n1/2*(v1 + v2);\n
LinearAlgebra[Norm](v1 + v2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
v1G-%'RTABLEG6$\"*'*H,X\"-%'MATRIXG6#7$7#\"\"#7#!\"%" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#v2G-%'RTABLEG6$\"*wI,X\"-%'MATRIXG6#7$7#\"\"&7#\"
\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*cJ,X\"-%'MATRIXG
6#7$7#\"\"(7#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*O
K,X\"-%'MATRIXG6#7$7#!\"$7#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
RTABLEG6$\"*;L,X\"-%'MATRIXG6#7$7##\"\"(\"\"#7##!\"\"F." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"#\"\"\"\"\"&" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 70 "The three dimensional coordinate \+
system and three dimensional vectors " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 74 "In a three dimensio
nal coordinate system, another axis, normally labelled " }{TEXT 328 1 
"z" }{TEXT -1 24 ", is added to the usual " }{TEXT 329 1 "x" }{TEXT 
-1 5 " and " }{TEXT 330 1 "y" }{TEXT -1 219 " axes of the standard two
 dimensional system, in order to provide a sense of a third dimension \+
of space measurement. The axes are understood to be mutually orthogona
l (with each axis at a right angles to the other two)." }}{PARA 0 "" 
0 "" {TEXT -1 129 "A three dimensional coordinate system is usually de
picted using what is called the \"right-hand rule\", and the system is
 called a " }{TEXT 259 30 "right-handed coordinate system" }{TEXT -1 
57 ". If the fingers of the right hand are curled around the " }{TEXT 
331 1 "z" }{TEXT -1 90 " axis, indicating the direction of a positive \+
rotation from the positive direction of the " }{TEXT 332 1 "x" }{TEXT 
-1 44 " axis towards the positive direction of the " }{TEXT 333 1 "y" 
}{TEXT -1 69 " axis, then the raised thumb points in the positive dire
ction of the " }{TEXT 334 1 "z" }{TEXT -1 128 " axis. Alternatively, i
f you hold up the thumb, index, and middle finger of the right hand, y
ou will see the orientation of the " }{TEXT 432 1 "x" }{TEXT -1 2 ", \+
" }{TEXT 433 1 "y" }{TEXT -1 6 ", and " }{TEXT 434 1 "z" }{TEXT -1 17 
" axes, where the " }{TEXT 261 12 "index finger" }{TEXT -1 42 " repres
ents the positive direction of the " }{TEXT 435 1 "x" }{TEXT 261 5 " a
xis" }{TEXT -1 6 ", the " }{TEXT 257 13 "middle finger" }{TEXT -1 42 "
 represents the positive direction of the " }{TEXT 436 1 "y" }{TEXT 
257 5 " axis" }{TEXT -1 9 " and the " }{TEXT 260 5 "thumb" }{TEXT -1 
42 " represents the positive direction of the " }{TEXT 437 1 "z" }
{TEXT 260 5 " axis" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
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ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cur
ve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14
" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "C
urve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The coordinates in a th
ree dimensional system are of the form" }{XPPEDIT 18 0 "``(x,y,z)" "6#
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"For example, the point " }{XPPEDIT 18 0 "P(1,3,2)" "6#-%\"PG6%\"\"\"
\"\"$\"\"#" }{TEXT -1 55 " is found by starting at the origin O (with \+
coordinates" }{XPPEDIT 18 0 " ``(0,0,0)" "6#-%!G6%\"\"!F&F&" }{TEXT 
-1 65 " ), moving a distance of 1 unit in the positive direction of th
e " }{TEXT 335 1 "x" }{TEXT -1 60 " axis, then moving 3 units in the p
ositive direction of the " }{TEXT 336 1 "y" }{TEXT -1 68 " axis, and  \+
finally moving 2 units in the positive direction of the " }{TEXT 438 
1 "z" }{TEXT -1 7 " axis. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
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{TEXT -1 4 "The " }{TEXT 439 1 "x" }{TEXT -1 5 " and " }{TEXT 440 1 "y
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h " }{XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"!" }{TEXT -1 41 ", that is, it i
s the plane with equation " }{XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"!" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "Similarly, the " }{TEXT 
442 2 "xz" }{TEXT -1 20 " plane has equation " }{XPPEDIT 18 0 "y=0" "6
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" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" }}}{PARA 256 "
" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 89 "The position vect
or of a point in 3 dimensions can be expressed using the three standar
d " }{TEXT 259 13 "basis vectors" }{TEXT -1 1 " " }{TEXT 337 1 "i" }
{TEXT -1 3 " , " }{TEXT 338 1 "j" }{TEXT -1 5 " and " }{TEXT 345 1 "k
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{TEXT 343 1 "i" }{TEXT -1 5 " + 3 " }{TEXT 344 1 "j" }{TEXT -1 5 " + 2
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\"\"$F%" }{TEXT -1 1 " " }{TEXT 408 1 "j" }{XPPEDIT 18 0 " ``- 2" "6#,
&%!G\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 427 1 "k" }{TEXT -1 2 ", \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 409 1 "v" }{XPPEDIT 18 0 "
``[1]+``" "6#,&&%!G6#\"\"\"F'F%F'" }{TEXT 410 1 "v" }{XPPEDIT 18 0 "``
[2] = 7;" "6#/&%!G6#\"\"#\"\"(" }{TEXT -1 1 " " }{TEXT 411 1 "i" }
{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " " }{TEXT 
412 1 "j" }{XPPEDIT 18 0 " ``+ 3" "6#,&%!G\"\"\"\"\"$F%" }{TEXT -1 1 "
 " }{TEXT 428 1 "k" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 413 1 "v" }{XPPEDIT 18 0 "``[1]-``;" "6#,&&%!G6#\"\"\"F'F%!\"
\"" }{TEXT 414 1 "v" }{XPPEDIT 18 0 "``[2] = -3;" "6#/&%!G6#\"\"#,$\"
\"$!\"\"" }{TEXT -1 1 " " }{TEXT 415 1 "i" }{XPPEDIT 18 0 "``-7;" "6#,
&%!G\"\"\"\"\"(!\"\"" }{TEXT -1 1 " " }{TEXT 416 1 "j" }{XPPEDIT 18 0 
"`` + 7" "6#,&%!G\"\"\"\"\"(F%" }{TEXT -1 1 " " }{TEXT 429 1 "k" }
{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2
" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 3 " ( " }{TEXT 417 1 "v" }
{XPPEDIT 18 0 "``[1]+``;" "6#,&&%!G6#\"\"\"F'F%F'" }{TEXT 418 1 "v" }
{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 2 " )" }{XPPEDIT 18 
0 " ``= 7/2" "6#/%!G*&\"\"(\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 
419 1 "i" }{XPPEDIT 18 0 "``-1/2;" "6#,&%!G\"\"\"*&F%F%\"\"#!\"\"F(" }
{TEXT -1 1 " " }{TEXT 420 1 "j" }{XPPEDIT 18 0 " ``+ 3/2 " "6#,&%!G\"
\"\"*&\"\"$F%\"\"#!\"\"F%" }{TEXT -1 1 " " }{TEXT 430 1 "k" }{TEXT -1 
3 ",  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
4 " || " }{TEXT 421 1 "v" }{XPPEDIT 18 0 "``[1]+``;" "6#,&&%!G6#\"\"\"
F'F%F'" }{TEXT 422 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }
{TEXT -1 6 " || = " }{XPPEDIT 18 0 "sqrt(49+1+9) = sqrt(59);" "6#/-%%s
qrtG6#,(\"#\\\"\"\"F)F)\"\"*F)-F%6#\"#f" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 
20 "row or column vector" }{TEXT -1 57 " notation in terms of componen
t may also be used, as in  " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{TEXT 423 1 "v" }{XPPEDIT 18 0 "``[1]=2" "6#/&%!G6#\"\"\"\"\"#" }
{TEXT -1 1 " " }{TEXT 424 1 "i" }{XPPEDIT 18 0 " ``- 4" "6#,&%!G\"\"\"
\"\"%!\"\"" }{TEXT -1 1 " " }{TEXT 425 1 "j" }{XPPEDIT 18 0 " ``+ 5" "
6#,&%!G\"\"\"\"\"&F%" }{TEXT -1 1 " " }{TEXT 431 1 "k" }{TEXT -1 3 " =
 " }{XPPEDIT 18 0 "[2, -4, 5] = matrix([[2], [-4], [5]]);" "6#/7%\"\"#
,$\"\"%!\"\"\"\"&-%'matrixG6#7%7#F%7#,$F'F(7#F)" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The above
 calculations can be performed in Maple using the \"old\" vector struc
ture as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 160 "v1 := vector([2,-4,5]);\nv2 := vector([5,3,-2
]);\nv1 + v2 = evalm(v1 + v2);\nv1 - v2 = evalm(v1 - v2);\n1/2*(v1 + v
2) = evalm(1/2*(v1 + v2));\nlinalg[norm](v1+v2,2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#v1G-%'vectorG6#7%\"\"#!\"%\"\"&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#v2G-%'vectorG6#7%\"\"&\"\"$!\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/,&%#v1G\"\"\"%#v2GF&-%'vectorG6#7%\"\"(!\"\"\"\"$" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%#v1G\"\"\"%#v2G!\"\"-%'vectorG6#7
%!\"$!\"(\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%#v1G#\"\"\"\"\"#
*&F&F'%#v2GF'F'-%'vectorG6#7%#\"\"(F(#!\"\"F(#\"\"$F(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"#f\"\"\"" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Alternatively, using the \"new
\" (column) Vector structure we have . . . " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "v1 := Vector([2,-
4,5]);\nv2 := Vector([5,3,-2]);\nv1 + v2;\nv1 - v2;\n1/2*(v1 + v2);\nL
inearAlgebra[Norm](v1 + v2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v
1G-%'RTABLEG6$\"*'R8]9-%'MATRIXG6#7%7#\"\"#7#!\"%7#\"\"&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#v2G-%'RTABLEG6$\"*OM,X\"-%'MATRIXG6#7%7#\"\"&
7#\"\"$7#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*;N,X\"
-%'MATRIXG6#7%7#\"\"(7#!\"\"7#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'RTABLEG6$\"*'f8]9-%'MATRIXG6#7%7#!\"$7#!\"(7#\"\"(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*wO,X\"-%'MATRIXG6#7%7##\"\"(\"\"#7#
#!\"\"F.7##\"\"$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"#f
\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 
5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 15 "For any vec
tor " }{TEXT 475 1 "v" }{XPPEDIT 18 0 "``[1]= x[1]" "6#/&%!G6#\"\"\"&%
\"xG6#F'" }{TEXT -1 1 " " }{TEXT 474 1 "i" }{XPPEDIT 18 0 " ``+ y[1]" 
"6#,&%!G\"\"\"&%\"yG6#F%F%" }{TEXT -1 1 " " }{TEXT 473 1 "j" }
{XPPEDIT 18 0 "`` + z[1]" "6#,&%!G\"\"\"&%\"zG6#F%F%" }{TEXT -1 1 " " 
}{TEXT 472 1 "k" }{TEXT -1 12 " and scalar " }{TEXT 476 1 "s" }{TEXT 
-1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 4 " || " }{TEXT 468 1 
"s" }{TEXT -1 1 " " }{TEXT 467 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#
\"\"\"" }{TEXT -1 9 " || = || " }{XPPEDIT 18 0 "s*x[1]" "6#*&%\"sG\"\"
\"&%\"xG6#F%F%" }{TEXT -1 1 " " }{TEXT 469 1 "i" }{XPPEDIT 18 0 " ``+ \+
s*y[1]" "6#,&%!G\"\"\"*&%\"sGF%&%\"yG6#F%F%F%" }{TEXT -1 1 " " }{TEXT 
470 1 "j" }{XPPEDIT 18 0 " ``+ s*z[1]" "6#,&%!G\"\"\"*&%\"sGF%&%\"zG6#
F%F%F%" }{TEXT -1 1 " " }{TEXT 471 1 "k" }{TEXT -1 6 " || = " }
{XPPEDIT 18 0 "sqrt(s^2*x[1]^2+s^2*y[1]^2+s^2*z[1]^2) = abs(s)*sqrt(x[
1]^2+y[1]^2+z[1]^2)" "6#/-%%sqrtG6#,(*&%\"sG\"\"#&%\"xG6#\"\"\"F*F.*&F
)F*&%\"yG6#F.F*F.*&F)F*&%\"zG6#F.F*F.*&-%$absG6#F)F.-F%6#,(*$&F,6#F.F*
F.*$&F16#F.F*F.*$&F56#F.F*F.F." }{XPPEDIT 18 0 " ``= abs(s)" "6#/%!G-%
$absG6#%\"sG" }{TEXT -1 4 " || " }{TEXT 477 1 "v" }{XPPEDIT 18 0 "``[1
]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " ||. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 15 "" 0 "" {TEXT -1 15 "For any vector " }{TEXT 481 1 "v" }
{XPPEDIT 18 0 "``[1]= x[1]" "6#/&%!G6#\"\"\"&%\"xG6#F'" }{TEXT -1 1 " \+
" }{TEXT 480 1 "i" }{XPPEDIT 18 0 " ``+ y[1]" "6#,&%!G\"\"\"&%\"yG6#F%
F%" }{TEXT -1 1 " " }{TEXT 479 1 "j" }{XPPEDIT 18 0 "`` + z[1]" "6#,&%
!G\"\"\"&%\"zG6#F%F%" }{TEXT -1 1 " " }{TEXT 478 1 "k" }{TEXT -1 17 " \+
we can obtain a " }{TEXT 259 11 "unit vector" }{TEXT -1 56 " (a vector
 of length 1) which has the same direction as " }{TEXT 573 1 "v" }
{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 22 " by taking the ve
ctor " }}{PARA 256 "" 0 "" {TEXT -1 7 "  ( || " }{TEXT 482 1 "v" }
{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " || )" }{XPPEDIT 
18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 483 1 "v" 
}{XPPEDIT 18 0 "``[1] = 1/sqrt(x[1]^2+y[1]^2+z[1]^2)" "6#/&%!G6#\"\"\"
*&F'F'-%%sqrtG6#,(*$&%\"xG6#F'\"\"#F'*$&%\"yG6#F'F1F'*$&%\"zG6#F'F1F'!
\"\"" }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }
{TEXT -1 1 " " }{TEXT 486 1 "i" }{XPPEDIT 18 0 " ``+ y[1]" "6#,&%!G\"
\"\"&%\"yG6#F%F%" }{TEXT -1 1 " " }{TEXT 485 1 "j" }{XPPEDIT 18 0 "`` \+
+ z[1]" "6#,&%!G\"\"\"&%\"zG6#F%F%" }{TEXT -1 1 " " }{TEXT 484 1 "k" }
{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``
= x[1]/sqrt(x[1]^2+y[1]^2+z[1]^2)" "6#/%!G*&&%\"xG6#\"\"\"F)-%%sqrtG6#
,(*$&F'6#F)\"\"#F)*$&%\"yG6#F)F1F)*$&%\"zG6#F)F1F)!\"\"" }{TEXT -1 1 "
 " }{TEXT 488 1 "i" }{XPPEDIT 18 0 "`` + y[1]/sqrt(x[1]^2+y[1]^2+z[1]^
2)" "6#,&%!G\"\"\"*&&%\"yG6#F%F%-%%sqrtG6#,(*$&%\"xG6#F%\"\"#F%*$&F(6#
F%F2F%*$&%\"zG6#F%F2F%!\"\"F%" }{TEXT -1 1 " " }{TEXT 489 1 "j" }
{XPPEDIT 18 0 "`` + z[1]/sqrt(x[1]^2+y[1]^2+z[1]^2)" "6#,&%!G\"\"\"*&&
%\"zG6#F%F%-%%sqrtG6#,(*$&%\"xG6#F%\"\"#F%*$&%\"yG6#F%F2F%*$&F(6#F%F2F
%!\"\"F%" }{TEXT -1 1 " " }{TEXT 487 1 "k" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "For example, a un
it vector in the direction of  " }{TEXT 490 1 "i" }{TEXT -1 3 " + " }
{TEXT 491 1 "j" }{TEXT -1 3 " + " }{TEXT 492 1 "k" }{TEXT -1 13 " is g
iven by " }{XPPEDIT 18 0 "1/sqrt(3)" "6#*&\"\"\"F$-%%sqrtG6#\"\"$!\"\"
" }{TEXT -1 1 " " }{TEXT 493 1 "i" }{XPPEDIT 18 0 " ``+1/sqrt(3)" "6#,
&%!G\"\"\"*&F%F%-%%sqrtG6#\"\"$!\"\"F%" }{TEXT -1 1 " " }{TEXT 494 1 "
j" }{XPPEDIT 18 0 "``+1/sqrt(3)" "6#,&%!G\"\"\"*&F%F%-%%sqrtG6#\"\"$!
\"\"F%" }{TEXT -1 1 " " }{TEXT 495 1 "k" }{TEXT -1 3 ".  " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 8 "Example " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}{PARA 0 "" 0 "" {TEXT -1 37 "Consider the unit cube with vertices " }
{XPPEDIT 18 0 "``(0,0,0), ``(1,0,0), ``(1,1,0), ``(0,1,0), ``(0,0,1), \+
``(1,0,1), ``(1,1,1), ``(0,1,1)" "6*-%!G6%\"\"!F&F&-F$6%\"\"\"F&F&-F$6
%F)F)F&-F$6%F&F)F&-F$6%F&F&F)-F$6%F)F&F)-F$6%F)F)F)-F$6%F&F)F)" }
{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 454 413 
413 {PLOTDATA 2 "6@-%'CURVESG6'7$7$$\"\"!F)F(7$F($\"\"%F)7%7$$!+++++v!
#6$\"++++?P!\"*F*7$$\"+++++vF1F2-%&STYLEG6#%,PATCHNOGRIDG-%'COLOURG6&%
$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-F$6'7$F'7$$!\"#F)FK7%7$$!+
\"*p'p!=F4$!+4I.8>F4FJ7$FQFOF8F<FC-F$6'7$F'7$F+F(7%7$F2F6FW7$F2F/F8F<F
C-F$6&7$F'7$F($\"#`!\"\"7%7$$!+++++lF1$\"++++N]F4Fhn7$$\"+++++lF1F`oF8
-F=6&F?F)F)F)-F$6&7$F'7$$!\"$F)F[p7%7$$!+f!QS!GF4$!+T>'f*GF4Fjo7$FapF_
pF8Feo-F$6&7$F'7$FinF(7%7$F`oFcoFgp7$F`oF^oF8Feo-F$6%7&FJF*FWFJFeo-%*L
INESTYLEGFE-F$6%7'F*7$F+F+7$$FFF)Feq7$FKFeqF*FeoF^q-F$6%7$FJFfqFeoF^q-
F$6%7$Fdq7$FeqFKFeoF^q-F$6%7&FJF]rFWFcqFeoF^q-F$6%7$F*7$$\"\"\"F)$F[oF
)FeoF^q-F$6%7$F'7$$\"3Immmmmmmm!#=F\\s-%&COLORG6&F?F)$\"\")F[oF)FC-F$6
%7$F[sFdqF_sF^q-%%TEXTG6&7$$!#EF[o$!#HF[oQ\"x6\"Feo-%%FONTG6$%*HELVETI
CAG\"#5-Fhs6&7$$\"#\\F[o$!#DFLQ\"yF`tFeoFat-Fhs6&7$F[uFitQ\"zF`tFeoFat
-Fhs6&7$$!$&>FL$!#AF[oQ\"AF`tFeoFat-Fhs6&7$$\"#TF[o$FLF[oQ\"BF`tFeoFat
-Fhs6&7$F_v$\"$:%FLQ\"CF`tFeoFat-Fhs6&7$$\"$0\"FL$!#7F[oQ\"MF`tFeoFat-
Fhs6&7$F_v$FfrF[oQ\"OF`tFeoFat-Fhs6&7$$\"#()FL$\"#oFLQ\"GF`tFeoFat-Fhs
6&7$$\"#FF[oFeqQ)P(1,1,1)F`tFeoFat-Fhs6&7$F_v$\"#=F[oQ\"kF`tF<-Fbt6%Fd
t%%BOLDGFet-Fhs6&7$$\"$v\"FLF_vQ\"jF`tF<Fhx-Fhs6&7$$!#lFL$!#&*FLQ\"iF`
tF<Fhx-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F`tF`z-Fbt6#%(DEFAULTG-%
%VIEWG6$FczFcz" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7
" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Cur
ve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 2
0" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "
Curve 27" }}}{PARA 257 "" 0 "" {TEXT -1 62 "Let ABC be the triangle ha
ving vertices with position vectors " }{TEXT 496 1 "i" }{TEXT -1 2 ", \+
" }{TEXT 497 1 "j" }{TEXT -1 5 " and " }{TEXT 498 1 "k" }{TEXT -1 15 "
 respectively. " }}{PARA 257 "" 0 "" {TEXT -1 69 "Let M be the mid-poi
nt of AB, so that CM is a median of triangle ABC." }}{PARA 257 "" 0 "
" {TEXT -1 81 "Let G be the point which is two-thirds of the way from \+
C to M along CM, that is, " }{TEXT 499 2 "CG" }{XPPEDIT 18 0 " ``= 2/3
" "6#/%!G*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 500 2 "CM" }
{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 28 "We find the position \+
vector " }{TEXT 531 2 "OG" }{TEXT -1 8 " of G.  " }}{PARA 0 "" 0 "" 
{TEXT -1 22 "M has position vector " }{TEXT 501 2 "OM" }{XPPEDIT 18 0 
" ``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 502 1 "
i" }{XPPEDIT 18 0 " ``+ 1/2" "6#,&%!G\"\"\"*&F%F%\"\"#!\"\"F%" }{TEXT 
-1 1 " " }{TEXT 503 1 "j" }{TEXT -1 8 ". Since " }{TEXT 504 2 "OC" }
{TEXT -1 3 " = " }{TEXT 505 1 "k" }{TEXT -1 2 ", " }{TEXT 506 2 "CM" }
{TEXT -1 3 " = " }{TEXT 507 2 "OM" }{XPPEDIT 18 0 " ``-`` " "6#,&%!G\"
\"\"F$!\"\"" }{TEXT 508 2 "OC" }{XPPEDIT 18 0 "``" "6#%!G" }{XPPEDIT 
18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 
509 1 "i" }{XPPEDIT 18 0 " ``+1/2" "6#,&%!G\"\"\"*&F%F%\"\"#!\"\"F%" }
{TEXT -1 1 " " }{TEXT 510 1 "j" }{XPPEDIT 18 0 " ``-`` " "6#,&%!G\"\"
\"F$!\"\"" }{TEXT 511 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 5 "Then " }{TEXT 512 2 "CG" }{XPPEDIT 18 0 " ``= 2/3" "6#/%!G*&\"\"
#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 513 2 "CM" }{XPPEDIT 18 0 " \+
``= 1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 514 1 "i
" }{XPPEDIT 18 0 " ``+ 1/3" "6#,&%!G\"\"\"*&F%F%\"\"$!\"\"F%" }{TEXT 
-1 1 " " }{TEXT 515 1 "j" }{XPPEDIT 18 0 " ``- 2/3" "6#,&%!G\"\"\"*&\"
\"#F%\"\"$!\"\"F)" }{TEXT -1 1 " " }{TEXT 516 1 "k" }{TEXT -1 5 ", so \+
" }{TEXT 517 2 "OG" }{TEXT -1 3 " = " }{TEXT 518 2 "OC" }{TEXT -1 3 " \+
+ " }{TEXT 519 2 "CG" }{TEXT -1 3 " = " }{TEXT 524 1 "k" }{TEXT -1 5 "
 + ( " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " 
}{TEXT 520 1 "i" }{XPPEDIT 18 0 " ``+ 1/3" "6#,&%!G\"\"\"*&F%F%\"\"$!
\"\"F%" }{TEXT -1 1 " " }{TEXT 521 1 "j" }{XPPEDIT 18 0 " ``- 2/3" "6#
,&%!G\"\"\"*&\"\"#F%\"\"$!\"\"F)" }{TEXT -1 1 " " }{TEXT 522 1 "k" }
{TEXT -1 2 " )" }{XPPEDIT 18 0 "``= 1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" 
}{TEXT -1 1 " " }{TEXT 525 1 "i" }{XPPEDIT 18 0 " ``+ 1/3" "6#,&%!G\"
\"\"*&F%F%\"\"$!\"\"F%" }{TEXT -1 1 " " }{TEXT 526 1 "j" }{XPPEDIT 18 
0 " ``+ 1/3" "6#,&%!G\"\"\"*&F%F%\"\"$!\"\"F%" }{TEXT -1 1 " " }{TEXT 
527 1 "k" }{TEXT -1 2 ". " }{TEXT 523 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 1 ":" }}{PARA 15 
"" 0 "" {TEXT -1 1 " " }{TEXT 532 2 "OG" }{XPPEDIT 18 0 "`` = 1/3" "6#
/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 533 2 "OP" }{XPPEDIT 
18 0 "`` = 1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 3 " ( " }{TEXT 
534 1 "i" }{TEXT -1 3 " + " }{TEXT 535 1 "j" }{TEXT -1 3 " + " }{TEXT 
536 1 "k" }{TEXT -1 49 " ), so G lies on the line OP where P is the po
int" }{XPPEDIT 18 0 " ``(1,1,1)" "6#-%!G6%\"\"\"F&F&" }{TEXT -1 22 " w
ith position vector " }{TEXT 528 1 "i" }{TEXT -1 3 " + " }{TEXT 529 1 
"j" }{TEXT -1 3 " + " }{TEXT 530 1 "k" }{TEXT -1 1 "." }}{PARA 15 "" 
0 "" {TEXT -1 3 "|| " }{TEXT 540 2 "OG" }{TEXT -1 3 " ||" }{XPPEDIT 
18 0 "`` = 1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 4 " || " }
{TEXT 537 1 "i" }{TEXT -1 3 " + " }{TEXT 538 1 "j" }{TEXT -1 3 " + " }
{TEXT 539 1 "k" }{TEXT -1 6 " || = " }{XPPEDIT 18 0 "``(1/3)*`.`*``(sq
rt(3)) = sqrt(3)/3;" "6#/*(-%!G6#*&\"\"\"F)\"\"$!\"\"F)%\".GF)-F&6#-%%
sqrtG6#F*F)*&-F06#F*F)F*F+" }{XPPEDIT 18 0 "``=1/sqrt(3)" "6#/%!G*&\"
\"\"F&-%%sqrtG6#\"\"$!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 "The dist
ance between two points and the equation of a sphere " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "The pos
ition vectors of two points  " }{XPPEDIT 18 0 "P[1](x[1],y[1],z[1]);" 
"6#-&%\"PG6#\"\"\"6%&%\"xG6#F'&%\"yG6#F'&%\"zG6#F'" }{TEXT -1 5 " and \+
" }{XPPEDIT 18 0 "P[2](x[2],y[2],z[2]);" "6#-&%\"PG6#\"\"#6%&%\"xG6#F'
&%\"yG6#F'&%\"zG6#F'" }{TEXT -1 5 ", are" }}{PARA 0 "" 0 "" {TEXT -1 
2 "  " }{TEXT 447 2 "OP" }{XPPEDIT 18 0 "``[1] = x[1]" "6#/&%!G6#\"\"
\"&%\"xG6#F'" }{TEXT -1 1 " " }{TEXT 448 1 "i" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "``+ y[1]" "6#,&%!G\"\"\"&%\"yG6#F%F%" }{TEXT -1 1 " " }
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CCCCCCCCD73FB99999999999A03FF333333333333A3FF4CCCCCCCCCCDA3FB999999999
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F4CCCCCCCCCCDC3FB99999999999A03FF333333333332F3FF4CCCCCCCCCCDB3FB99999
999999A03FF333333333332C3FF4CCCCCCCCCCDA3FB99999999999A03FF33333333333
293FF4CCCCCCCCCCD73FB99999999999A03FF33333333333263FF4CCCCCCCCCCD43FB9
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3333243FF4CCCCCCCCCCCD3FB99999999999A03FF33333333333253FF4CCCCCCCCCCC9
3FB99999999999A03FF33333333333263FF4CCCCCCCCCCC63FB99999999999A03FF333
33333333293FF4CCCCCCCCCCC33FB99999999999A03FF333333333332C3FF4CCCCCCCC
CCC03FB99999999999A03FF333333333332F3FF4CCCCCCCCCCBF3FB99999999999A03F
F33333333333333FF4CCCCCCCCCCBE3FB99999999999A03FF33333333333373FF4CCCC
CCCCCCBF3FB99999999999A03FF333333333333A3FF4CCCCCCCCCCC03FB99999999999
A03FF333333333333D3FF4CCCCCCCCCCC33FB99999999999A03FF33333333333403FF4
CCCCCCCCCCC63FB99999999999A03FF33333333333413FF4CCCCCCCCCCC93FB9999999
9999A03FF33333333333423FF4CCCCCCCCCCCD3FB99999999999A0-%&COLORG6&FK$\"
#$*!\"#$\"\"*F*Faq-%*LINESTYLEG6#\"\"#-FE6#%&PATCHG-%%TEXTG6%7%$\"#BF*
FO$F`qF*Q\"xFjpFbo-F[r6%7%FOF^rF`rQ\"yFjpFbo-F[r6%7%FOF`r$\"#AF*Q\"zFj
pFbo-F[r6%7%F($\"#@F*F^rQ0centre:~(a,b,c)FjpFbo-F[r6%7%F(F_sF_sQ*radiu
s:~rFjpFbo-%+LIGHTMODELG6#%(LIGHT_4G-%*AXESSTYLEG6#%%NONEG-%+PROJECTIO
NG6%$\"#DFP$\"#qFPFT" 1 2 0 1 10 0 2 1 6 1 2 1.000000 70.000000 
25.000000 1 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}}{PARA 0 "" 0 "" {TEXT 
-1 63 "In particular, the sphere with centre at the origin and radius \+
" }{TEXT 466 1 "r" }{TEXT -1 14 " has equation " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2+z^2=r^2" "6#/,(*$%\"xG\"\"#\"\"
\"*$%\"yGF'F(*$%\"zGF'F(*$%\"rGF'" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}
{PARA 0 "" 0 "" {TEXT -1 7 "Given  " }{TEXT 541 1 "v" }{XPPEDIT 18 0 "
``[1] = 3;" "6#/&%!G6#\"\"\"\"\"$" }{TEXT -1 1 " " }{TEXT 542 1 "i" }
{XPPEDIT 18 0 "``+2;" "6#,&%!G\"\"\"\"\"#F%" }{TEXT -1 1 " " }{TEXT 
543 1 "j" }{XPPEDIT 18 0 "``-4;" "6#,&%!G\"\"\"\"\"%!\"\"" }{TEXT -1 
1 " " }{TEXT 555 1 "k" }{TEXT -1 5 " and " }{TEXT 544 1 "v" }{XPPEDIT 
18 0 "``[2]=`` " "6#/&%!G6#\"\"#F%" }{TEXT 545 1 "i" }{XPPEDIT 18 0 "`
`-5;" "6#,&%!G\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 546 1 "j" }
{XPPEDIT 18 0 "``+2;" "6#,&%!G\"\"\"\"\"#F%" }{TEXT -1 1 " " }{TEXT 
556 1 "k" }{TEXT -1 8 ", find: " }}{PARA 0 "" 0 "" {TEXT -1 6 " (a)  \+
" }{TEXT 547 1 "v" }{XPPEDIT 18 0 "``[1]+``" "6#,&&%!G6#\"\"\"F'F%F'" 
}{TEXT 548 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 9 "
,   (b)  " }{TEXT 549 1 "v" }{XPPEDIT 18 0 "``[1]-``;" "6#,&&%!G6#\"\"
\"F'F%!\"\"" }{TEXT 550 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" 
}{TEXT -1 9 ",   (c)  " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" 
}{TEXT -1 3 " ( " }{TEXT 551 1 "v" }{XPPEDIT 18 0 "``[1]+``;" "6#,&&%!
G6#\"\"\"F'F%F'" }{TEXT 552 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"
\"#" }{TEXT -1 14 " ),   (d)  || " }{TEXT 553 1 "v" }{XPPEDIT 18 0 "``
[1]+``;" "6#,&&%!G6#\"\"\"F'F%F'" }{TEXT 554 1 "v" }{XPPEDIT 18 0 "``[
2];" "6#&%!G6#\"\"#" }{TEXT -1 5 " ||, " }}{PARA 0 "" 0 "" {TEXT -1 
40 " (e) a unit vector in the direction of  " }{TEXT 557 1 "v" }
{XPPEDIT 18 0 "``[1]+``" "6#,&&%!G6#\"\"\"F'F%F'" }{TEXT 558 1 "v" }
{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " . " }}{PARA 0 "" 
0 "" {TEXT -1 44 "____________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 44 "____________________________________________" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q
2 " }}{PARA 0 "" 0 "" {TEXT -1 48 " This question is concerned with th
e two points " }{XPPEDIT 18 0 "A(3,-1,2)" "6#-%\"AG6%\"\"$,$\"\"\"!\"
\"\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B(6,4,8)" "6#-%\"BG6%\"\"'\"
\"%\"\")" }{TEXT -1 26 " and the origin O(0,0,0). " }}{PARA 0 "" 0 "" 
{TEXT -1 34 "(a) Find the position vectors (i) " }{TEXT 561 2 "OA" }
{TEXT -1 12 " of A, (ii) " }{TEXT 562 2 "OB" }{TEXT -1 13 " of B, (iii
) " }{TEXT 563 2 "OM" }{TEXT -1 45 " of the mid-point M of AB, in term
s of their " }{TEXT 564 1 "i" }{TEXT -1 2 ", " }{TEXT 565 1 "j" }
{TEXT -1 5 " and " }{TEXT 566 1 "k" }{TEXT -1 13 " components. " }}
{PARA 0 "" 0 "" {TEXT -1 24 "(b) Express the vectors " }{TEXT 559 2 "A
B" }{TEXT -1 2 ", " }{TEXT 560 2 "BA" }{TEXT -1 5 " and " }{TEXT 567 
2 "AM" }{TEXT -1 19 " in terms of their " }{TEXT 568 1 "i" }{TEXT -1 
2 ", " }{TEXT 569 1 "j" }{TEXT -1 5 " and " }{TEXT 570 1 "k" }{TEXT 
-1 14 " components.  " }}{PARA 0 "" 0 "" {TEXT -1 12 "(c) Find || " }
{TEXT 571 2 "AB" }{TEXT -1 5 " ||. " }}{PARA 0 "" 0 "" {TEXT -1 43 "(d
) Find a unit vector in the direction of " }{TEXT 572 2 "AB" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "_____________________________
_______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 44 "____________________________________________" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 
"" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 54 "The line which pass
es through the origin and the point" }{XPPEDIT 18 0 " ``(1,2,2)" "6#-%
!G6%\"\"\"\"\"#F'" }{TEXT -1 14 " has equation " }{XPPEDIT 18 0 "2*x=y
" "6#/*&\"\"#\"\"\"%\"xGF&%\"yG" }{XPPEDIT 18 0 "``=z" "6#/%!G%\"zG" }
{TEXT -1 56 ", while the sphere with centre at the origin and radius \+
" }{XPPEDIT 18 0 "3*sqrt(2)" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" }
{TEXT -1 14 " has equation " }{XPPEDIT 18 0 "x^2+y^2+z^2 = 18" "6#/,(*
$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'F(\"#=" }{TEXT -1 69 ". Find the \+
coordinates of the points where the line meets the sphere." }}{PARA 0 
"" 0 "" {TEXT -1 44 "____________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 44 "____________________________________________" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pi
ctures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT 260 13 "plot2dvector " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 993 "plot2dvector := p
roc(point::\{Vector, vector, list\}, vect::\{Vector, vector, list\},he
ad_width,head_relative_height)\n   local a, b, i, x, y, Cos, Sin, v, l
ocopts, L;\n   x := point[1];\n   y := point[2];\n   a := vect[1];\n  \+
 b := vect[2];\n   if type(vect,'list') then \n      return procname(p
oint,Vector(2,[a-x,b-y]),\n        head_width,head_relative_height,arg
s[5..-1])\n   end if;\n   L := evalf(sqrt(a^2+b^2));\n   if a=0 and b=
0 then return CURVES([]) end if;\n   Cos := evalf(a/L);\n   Sin := eva
lf(b/L);\n   v := [[[x,y],[x+a,y+b]],[[x-1/2*head_width*Sin \n      -h
ead_relative_height*Cos*L+1/2*a,\n        y+1/2*head_width*Cos-head_re
lative_height*Sin*L+1/2*b],\n      [x+1/2*a,y+1/2*b],  \n        [x+1/
2*head_width*Sin-head_relative_height*Cos*L+1/2*a, \n        y-1/2*hea
d_width*Cos -head_relative_height*Sin*L+1/2*b]]];\n    v := evalf(v);
\n    locopts := [args[5 .. -1]];\n    locopts := \n       convert(['s
tyle'='patchnogrid',op(locopts)],'PLOToptions');\n    PLOT(CURVES(op(v
),op(locopts)))\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" 
{TEXT -1 15 "basis vectors  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 919 "p1 := plottools[arrow]([0,0
],[3,2],.1,.08,.04,\n        arrow,color=COLOR(RGB,.85,0,.85),thicknes
s=2):\np2 := plottools[arrow]([0,0],[1,0],.1,.08,.08,\n               \+
         arrow,color=red,thickness=3):\np3 := plottools[arrow]([0,0],[
0,1],.1,.08,.08,\n                        arrow,color=red,thickness=3)
:\np4 := plot([[[3,2]]$3],style=point,color=black,\n                  \+
 symbol=[circle,cross,diamond]):\nt1 := plots[textplot]([[.5,-.2,`i`],
[-.2,.5,`j`]],\n                    color=red,font=[HELVETICA,BOLD,10]
):\nt2 := plots[textplot]([[1.85,.9,`v =   i      j`]],\n        color
=COLOR(RGB,.85,0,.85),font=[HELVETICA,BOLD,10]):\nt3 := plots[textplot
]([[1.95,.9,`3  + 2`]],\n        color=COLOR(RGB,.85,0,.85),font=[HELV
ETICA,10]):\nt4 := plots[textplot]([[3.2,2.2,`P( 3,2 )`]],\n        co
lor=black,font=[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,t1,t2,t3,
t4],labels=[`x`,`y`],\n        tickmarks=[5,4],view=[-.3..4,-.3..3]);
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "addition o
f vectors  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1889 "p1 := plottools[arrow]([0,0],[5,2],.1,.17,.05,\n \+
       arrow,color=COLOR(RGB,.85,0,.85)):\np2 := plottools[arrow]([5,2
],[8,8],.1,.17,.05,\n                arrow,color=COLOR(RGB,0,.8,0)):\n
p3 := plottools[arrow]([0,0],[8,8],.1,.17,.04,\n                      \+
  arrow,color=brown):\np4 := plottools[arrow]([0,0],[5,0],.1,.17,.05,
\n                        arrow,color=red):\np5 := plottools[arrow]([5
,0],[5,2],.1,.13,.15,\n                  arrow,color=COLOR(RGB,.5,0,1)
):\np6 := plottools[arrow]([5,2],[8,2],.1,.17,.05,\n                  \+
      arrow,color=red):\np7 := plottools[arrow]([8,2],[8,8],.1,.17,.05
,\n                 arrow,color=COLOR(RGB,.5,0,1)):\nt1 := plots[textp
lot]([[3.33,4.23,`v   + v`]],\n     color=brown,font=[HELVETICA,BOLD,1
0]):\nt2 := plots[textplot]([[3.5,4.1,`1         2`]],\n     color=bro
wn,font=[HELVETICA,8]):\nt3 := plots[textplot]([[3,1,`v`]],\n   color=
COLOR(RGB,.85,0,.85),font=[HELVETICA,BOLD,10]):\nt4 := plots[textplot]
([[3.17,.87,`1`]],\n   color=COLOR(RGB,.85,0,.85),font=[HELVETICA,8]):
\nt5 := plots[textplot]([[6.5,4.6,`v`]],\n   color=COLOR(RGB,0,.8,0),f
ont=[HELVETICA,BOLD,10]):\nt6 := plots[textplot]([[6.67,4.47,`2`]],\n \+
  color=COLOR(RGB,0,.8,0),font=[HELVETICA,8]):\nt7 := plots[textplot](
[[3,-.25,`i`],[6.5,1.75,`i`]],\n       color=red,font=[HELVETICA,BOLD,
10]):\nt8 := plots[textplot]([[2.7,-.25,`x`],[6.2,1.75,`x`]],\n       \+
color=red,font=[HELVETICA,10]):\nt9 := plots[textplot]([[2.85,-.4,`1`]
,[6.35,1.6,`2`]],\n       color=red,font=[HELVETICA,8]):\nt10 := plots
[textplot]([[5.45,1,`j`],[8.45,4.6,`j`]],\n     color=COLOR(RGB,.5,0,1
),font=[HELVETICA,BOLD,10]):\nt11 := plots[textplot]([[5.15,1,`y`],[8.
15,4.6,`y`]],\n     color=COLOR(RGB,.5,0,1),font=[HELVETICA,10]):\nt12
 := plots[textplot]([[5.3,.85,`1`],[8.3,4.45,`2`]],\n     color=COLOR(
RGB,.5,0,1),font=[HELVETICA,8]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7
,\n      t1,t2,t3,t4,t5,t6,t7,t8,t9,t10,t11,t12],axes=none);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "right-handed coord syst
em   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1827 "d := evalf(Pi/20):\np1 := plot([1/16*y^2-5/8*y-55/1
6,y,y=1..9],color=brown):\np2 := plot(x^2/16,x=-4..4,color=brown):\np3
 := plot(1/19*x^2+3/76*x+2,x=-19/4..4,color=brown):\np4 := plot(1/20*x
^2+1/20*x+4,x=-5..4,color=brown):\np5 := plot(1/19*x^2+3/76*x+6,x=-19/
4..4,color=brown):\np6 := plot(1/12*x^2+1/12*x+8,x=-4..3,color=brown):
\np7 := plot(-1/8*x^2+11,x=-4..0,color=brown):\np8 := plot([1/16*y^2-1
5/8*y+209/16,y,y=11..15],color=brown):\np9 := plot(-1/360*x^3+29/180*x
^2-313/120*x+349/20,\n                   x=1..9,color=brown):\np10 := \+
plot(1/21*x^2-5/21*x-5/7,\n                   x=-4..9,color=brown):\np
11 := plot([-1/8*y^2+3/4*y+67/8,y,y=1..5],color=brown):\np12 := plot([
seq([cos(d*i),15+sin(i*d)],i=0..20)],color=brown):\np13 := plot([-1/2*
y^2+2*y+5/2,y,y=1..3],color=brown):\np14 := plot([-1/2*y^2+4*y-7/2,y,y
=3..5],color=brown):\np15 := plot([-1/2*y^2+6*y-27/2,y,y=5..7],color=b
rown):\np16 := plot([-35/132*y^2+529/132*y-255/22,y,y=34/5..9],color=b
rown):\nP := _u -> op(op(1,op(1,_u))):\nR := _u -> op(ListTools[Revers
e](_u)):\np17 := plots[polygonplot]([P(p8),R([P(p12)]),P(p9),R([P(p11)
]),\n       R([P(p10)]),P(p1),P(p7)],\n   color=COLOR(RGB,1,.85,.75),s
tyle=patchnogrid):\np18 := plottools[arrow]([0,8],[0,30],.1,1,.05,\n  \+
             arrow,color=black):\np19 := plot([[0,-15],[0,0]],color=bl
ack):\np20 := plottools[arrow]([0,-15],[30,-15],.1,1,.03,\n           \+
    arrow,color=black):\np21 := plottools[arrow]([0,-15],[-20,-35],.1,
1,.05,\n               arrow,color=black):\np22 := plots[polygonplot](
[[0,-15],[28,-15],[10,-33],[-18,-33]],\n   style=patchnogrid,color=COL
OR(RGB,.93,.93,.93)):\nt1 := plots[textplot]([[-18,-36,`x`],[29,-17,`y
`],\n [-1.5,28,`z`]],color=black,font=[HELVETICA,10]):\nplots[display]
([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,\n      p12,p13,p14,p15,p16,p17,p
18,p19,p20,p21,p22,t1],axes=none,\n    view=[-20..31,-36..30]);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "interp([-19/4,0,4],[3,2,3],x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "interp([1,5,9],[-4,-5,-4],7);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "interp([-5,0,4],[5,4,5],x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "interp([-4,0,4],[1,0,1],x);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "interp([-19/4,0,4],[7,6
,7],x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "interp([-4,0,3],
[9,8,9],x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "interp([-4,-
2,0],[9,21/2,11],x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int
erp([11,13,15],[0,-3/4,-1],y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "interp([1,3,6,9],[15,11,7,5],x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "interp([-4,3,9],[1,-1,1],x);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "interp([1,3,5],[9,19/2,9],y);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "interp([1,2,3],[4,9/2,4],y);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "interp([3,4,5],[4,9/2,4],y);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "interp([5,6,7],[4,9/2,4],y);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "interp([34/5,8,9],[17/5
,7/2,3],y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 23 "plottin
g a point in 3d " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1180 "p1 := plottools[arrow]([-5,0],[16,0],.1,.4,.
03,\n                    arrow,color=black):\np2 := plottools[arrow]([
0,-5],[0,12],.1,.4,.03,\n                    arrow,color=black):\np3 :
= plottools[arrow]([4,4],[-8,-8],.1,.4,.03,\n                    arrow
,color=black):\np4 := plot([[[-3,-3],[9,-3],[12,0]],[[0,8],[-3,5],\n  \+
 [9,5],[12,8],[0,8]],[[-3,-3],[-3,5]],[[9,-3],[9,5]],\n  [[12,0],[12,8
]]],color=black,linestyle=2):\np5 := plot([[[-4,-.3],[-4,.3]],[[4,-.3]
,[4,.3]],[[8,-.3],[8,.3]],\n     [[12,-.3],[12,.3]],[[-6,-6.3],[-6,-5.
7]],\n   [[-3,-3.3],[-3,-2.7]],[[3,3.3],[3,2.7]],[[-.2,-4],[.2,-4]],\n
   [[-.2,4],[.2,4]],[[-.2,8],[.2,8]]],color=black):\np6 := plot([[[9,5
]]$4],style=point,symbolsize=[12$3,10],\n       symbol=[circle,diamond
,cross,diamond],color=red):\nt1 := plots[textplot]([[15.5,-.7,`y`],[-.
6,11.5,`z`],\n     [-7.6,-8.3,`x`],[11,5,`P(1,3,2)`]],\n        color=
black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-4,-.7,`-1`],[4,-
.7,`1`],[8,-.7,`2`],\n      [12,-.7,`3`],[-.6,-4,`-1`],[-.6,4,`1`],[-.
6,8,`2`],\n     [3.1,2.4,`-1`],[-2.9,-3.7,`1`],[-5.9,-6.7,`2`]],\n    \+
    color=COLOR(RGB,.01,.01,.01),font=[HELVETICA,8]):\nplots[display](
[p1,p2,p3,p4,p5,p6,t1,t2],axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "
" 0 "" {TEXT -1 23 "the coordinate planes  " }}{PARA 0 "" 0 "" {TEXT 
-1 8 "? scrap " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1103 "p1 := pl
ottools[arrow]([0,0,0],[1.2,0,0],.1,.05,.05,\n                    arro
w,color=black):\np2 := plottools[arrow]([0,0,0],[0,1.2,0],.1,.05,.05,
\n                    arrow,color=black):\np3 := plottools[arrow]([0,0
,0],[0,0,1.2],.1,.05,.05,\n                    arrow,color=black):\np4
 := plots[polygonplot3d]([[0,0,0],[0,0,1],[0,1,1],\n           [0,1,0]
],style=patchnogrid,color=COLOR(RGB,.9,.9,.97)):\np5 := plots[polygonp
lot3d]([[0,0,0],[0,0,1],[1,0,1],\n           [1,0,0]],style=patchnogri
d,color=COLOR(RGB,.9,.97,.9)):\np6 := plots[polygonplot3d]([[0,0,0],[1
,0,0],[1,1,0],\n           [0,1,0]],style=patchnogrid,color=COLOR(RGB,
.97,.9,.9)):\nt1 := plots[textplot3d]([[1.1,0,-.1,`x`],[0,1.1,-.1,`y`]
,\n       [0,-.1,1.1,`z`]],\n   color=black,font=[HELVETICA,10]):\nt2 \+
:= plots[textplot3d]([[-.05,1.05,1.05,`yz plane`]],\n   color=blue,fon
t=[HELVETICA,10]):\nt3 := plots[textplot3d]([[1.05,1.05,-.05,`xy plane
`]],\n   color=red,font=[HELVETICA,10]):\nt4 := plots[textplot3d]([[1.
05,-.05,1.05,`xz plane`]],\n   color=COLOR(RGB,0,.8,0),font=[HELVETICA
,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,t1,t2,t3,t4],axes=none); " }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1504 "p1 := plottools[arrow]([0,
0,0],[1.2,0,0],.1,.05,.05,\n                    arrow,color=black):\np
2 := plottools[arrow]([0,0,0],[0,1.2,0],.1,.05,.05,\n                 \+
   arrow,color=black):\np3 := plottools[arrow]([0,0,0],[0,0,1.2],.1,.0
5,.05,\n                    arrow,color=black):\np4 := plot3d([s,t,0],
s=0.01..0.99,t=0.01..0.99,gridstyle=triangular,\n      grid=[16,16],st
yle=wireframe,color=COLOR(RGB,1,.85,.85)):\np5 := plot3d([s,0,t],s=0.0
1..0.99,t=0.01..0.99,gridstyle=triangular,\n      grid=[16,16],style=w
ireframe,color=COLOR(RGB,.85,1,.85)):\np6 := plot3d([0,s,t],s=0.01..0.
99,t=0.01..0.99,gridstyle=triangular,\n      grid=[16,16],style=wirefr
ame,color=COLOR(RGB,.85,.85,1)):\np44 := plots[polygonplot3d]([[0,0,0]
,[0,0,1],[0,1,1],\n           [0,1,0]],style=patchnogrid,color=COLOR(R
GB,.9,.9,.97)):\np55 := plots[polygonplot3d]([[0,0,0],[0,0,1],[1,0,1],
\n           [1,0,0]],style=patchnogrid,color=COLOR(RGB,.9,.97,.9)):\n
p66 := plots[polygonplot3d]([[0,0,0],[1,0,0],[1,1,0],\n           [0,1
,0]],style=patchnogrid,color=COLOR(RGB,.97,.9,.9)):\nt1 := plots[textp
lot3d]([[1.1,0,-.1,`x`],[0,1.1,-.1,`y`],\n       [0,-.1,1.1,`z`]],\n  \+
 color=black,font=[HELVETICA,10]):\nt2 := plots[textplot3d]([[-.05,1.0
5,1.05,`yz plane`]],\n   color=blue,font=[HELVETICA,10]):\nt3 := plots
[textplot3d]([[1.05,1.05,-.05,`xy plane`]],\n   color=red,font=[HELVET
ICA,10]):\nt4 := plots[textplot3d]([[1.05,-.05,1.05,`xz plane`]],\n   \+
color=COLOR(RGB,0,.8,0),font=[HELVETICA,10]):\nplots[display]([p1,p2,p
3,p4,p5,p6,t1,t2,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 
0 "" {TEXT -1 42 "3d basis vectors and position vectors - 2d" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1933 "
p1 := plottools[arrow]([-5,0],[16,0],.1,.4,.03,\n                    a
rrow,color=black):\np2 := plottools[arrow]([0,-5],[0,12],.1,.4,.03,\n \+
                   arrow,color=black):\np3 := plottools[arrow]([4,4],[
-8,-8],.1,.4,.03,\n                    arrow,color=black):\np4 := plot
tools[arrow]([0,0],[4,0],.1,.4,.07,\n                    arrow,color=r
ed,thickness=2):\np5 := plottools[arrow]([0,0],[0,4],.1,.4,.07,\n     \+
               arrow,color=red,thickness=2):\np6 := plottools[arrow]([
0,0],[-3,-3],.1,.4,.07,\n                    arrow,color=red,thickness
=2):\np7 := plottools[arrow]([0,0],[9,5],.1,.4,.05,\n         arrow,co
lor=COLOR(RGB,.85,0,.85),thickness=2):\np8 := plot([[[-3,-3],[9,-3],[1
2,0]],[[0,8],[-3,5],\n   [9,5],[12,8],[0,8]],[[-3,-3],[-3,5]],[[9,-3],
[9,5]],\n    [[12,0],[12,8]]],color=black,linestyle=2):\np9 := plot([[
[-4,-.3],[-4,.3]],[[4,-.3],[4,.3]],[[8,-.3],[8,.3]],\n     [[12,-.3],[
12,.3]],[[-6,-6.3],[-6,-5.7]],\n   [[-3,-3.3],[-3,-2.7]],[[3,3.3],[3,2
.7]],[[-.2,-4],[.2,-4]],\n   [[-.2,4],[.2,4]],[[-.2,8],[.2,8]]],color=
black):\np10 := plot([[[9,5]]$3],style=point,symbol=[circle,diamond,cr
oss],\n            color=brown):\nt1 := plots[textplot]([[15.5,-.7,`y`
],[-.6,11.5,`z`],\n     [-7.6,-8.3,`x`],[10.5,5,`P(1,3,2)`]],\n       \+
 color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-4,-.7,`-1
`],[4,-.7,`1`],[8,-.7,`2`],\n      [12,-.7,`3`],[-.6,-4,`-1`],[-.6,4,`
1`],[-.6,8,`2`],\n     [3.1,2.4,`-1`],[-2.9,-3.7,`1`],[-5.9,-6.7,`2`]]
,\n       color=COLOR(RGB,.01,.01,.01),font=[HELVETICA,8]):\nt3 := plo
ts[textplot]([[-1.1,-1.7,`i`],[2,-.7,`j`],[-.5,2,`k`]],\n        color
=red,font=[HELVETICA,BOLD,10]):\nt4 := plots[textplot]([7.4,2.8,`i    \+
  j      k`],\n        color=COLOR(RGB,.85,0,.85),font=[HELVETICA,BOLD
,10]):\nt5 := plots[textplot]([7.4,2.8,`+ 3  + 2`],\n        color=COL
OR(RGB,.85,0,.85),font=[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,p
5,p6,p7,p8,p9,p10,\n                                t1,t2,t3,t4,t5],ax
es=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 42 "3d basis
 vectors and position vectors - 3d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2610 "p1 := plottools[arrow]([-5
,0,0],[8,0,0],.1,.4,.04,\n                    arrow,color=black):\np2 \+
:= plottools[arrow]([0,-5,0],[0,16,0],.1,.4,.04,\n                    \+
arrow,color=black):\np3 := plottools[arrow]([0,0,-5],[0,0,12],.1,.4,.0
4,\n                    arrow,color=black):\np4 := plottools[arrow]([0
,0,0],[4,0,0],.1,.4,.07,\n                    arrow,color=red,thicknes
s=2):\np5 := plottools[arrow]([0,0,0],[0,4,0],.1,.4,.07,\n            \+
        arrow,color=red,thickness=2):\np6 := plottools[arrow]([0,0,0],
[0,0,4],.1,.4,.07,\n                    arrow,color=red,thickness=2):
\np7 := plottools[arrow]([0,0,0],[4,12,8],.1,.4,.05,\n         arrow,c
olor=COLOR(RGB,.85,0,.85),thickness=2):\np8 := plots[pointplot3d]([[0,
12,8],[0,0,8],[4,0,8],[4,12,8]],\n                    style=line,color
=black,linestyle=2):\np9 := plots[pointplot3d]([[4,0,8],[4,0,0],[4,12,
0],[4,12,8],\n    [0,12,8],[0,12,0],[4,12,0]],style=line,color=black,l
inestyle=2):\np10 := plots[pointplot3d]([[-4,0,-.3],[-4,0,.3]],style=l
ine,color=black):\np11 := plots[pointplot3d]([[4,0,-.3],[4,0,.3]],styl
e=line,color=black):\np12 := plots[pointplot3d]([[0,-4,-.3],[0,-4,.3]]
,style=line,color=black):\np13 := plots[pointplot3d]([[0,4,-.3],[0,4,.
3]],style=line,color=black):\np14 := plots[pointplot3d]([[0,8,-.3],[0,
8,.3]],style=line,color=black):\np15 := plots[pointplot3d]([[0,12,-.3]
,[0,12,.3]],style=line,color=black):\np16 := plots[pointplot3d]([[0,-.
3,-4],[0,.3,-4]],style=line,color=black):\np17 := plots[pointplot3d]([
[0,-.3,4],[0,.3,4]],style=line,color=black):\np18 := plots[pointplot3d
]([[0,-.3,8],[0,.3,8]],style=line,color=black):\np19 := plots[pointplo
t3d]([[4,12,8]],style=point,\n                symbol=circle,symbolsize
=8,color=brown):\nt1 := plots[textplot3d]([[0,15.5,-1,`y`],[0,-1,11.5,
`z`],\n     [7.5,0,-1,`x`],[4,13,8,`P(1,3,2)`]],\n        color=COLOR(
RGB,.01,.01,.01),font=[HELVETICA,10]):\nt2 := plots[textplot3d]([[0,-4
.2,-.9,`-1`],[0,3.8,-.9,`1`],[0,7.8,-.9,`2`],\n      [0,11.8,-.9,`3`],
[-4,0,-.9,`-1`],[4,0,-.9,`1`],\n     [0,-.9,-4.2,`-1`],[0,-.7,3.8,`1`]
,[0,-.7,7.8,`2`]],\n       color=COLOR(RGB,.01,.01,.01),font=[HELVETIC
A,8]):\nt3 := plots[textplot3d]([[2,0,-.9,`i`],[0,2,-.9,`j`],[0,-.9,2,
`k`]],\n        color=red,font=[HELVETICA,BOLD,10]):\nt4 := plots[text
plot3d]([2,6.5,3.4,`i      j      k`],\n        color=COLOR(RGB,.85,0,
.85),font=[HELVETICA,BOLD,10]):\nt5 := plots[textplot3d]([2,6.5,3.4,` \+
 + 3  + 2`],\n        color=COLOR(RGB,.85,0,.85),font=[HELVETICA,10]):
\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,\n       \+
   p14,p15,p16,p17,p18,p19,t1,t2,t3,t4,t5],axes=none,\n     view=[-5..
8,-5..16,-5..12],orientation=[23,61]); " }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 
0 "" 0 "" {TEXT -1 23 "length of a 3d vector  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1118 "p1 := plot
tools[arrow]([0,0],[5,4],.1,.23,.05,\n                 arrow,color=bro
wn,thickness=2):\np2 := plottools[arrow]([-.5,0],[9,0],.1,.2,.03,\n   \+
              arrow,color=black):\np3 := plottools[arrow]([0,-.3],[0,9
],.1,.2,.05,\n                 arrow,color=black):\np4 := plottools[ar
row]([.2,.3],[-3,-4.5],.1,.2,.06,\n                 arrow,color=black)
:\np5 := plot([[[5,4],[-2,-3],[5,-3],[7,0],[5,4],[5,-3],[0,0]],\n     \+
 [[-2,-3],[5,4],[0,7]]],linestyle=2,color=black):\nt1 := plots[textplo
t]([[-2.65,-4.7,`x`],[8.6,-.4,`y`],\n       [-.3,8.6,`z`],[-1.9,-3.3,`
x`],[7,-.5,`y`],[-.4,7,`z`],\n       [-.3,.5,`O`],[5.2,-3.3,`Q`],[6.1,
4.2,`P(x ,y ,z  )`]],\n        color=black,font=[HELVETICA,10]):\nt2 :
= plots[textplot]([[-1.9,-3.6,`    1`],[7,-.8,`    1`],\n       [-.4,6
.7,`    1`],[6.1,3.9,`     1   1   1`]],\n        color=black,font=[HE
LVETICA,8]):\nt3 := plots[textplot]([2,2.4,`v`],color=brown,\n        \+
                           font=[HELVETICA,BOLD,10]):\nt4 := plots[tex
tplot]([2,2.1,`   1`],color=brown,\n                                  \+
 font=[HELVETICA,8]):\nplots[display]([p1,p2,p3,p4,p5,t1,t2,t3,t4],axe
s=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 26 "distance
 between 2 points " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 799 "p1 := plot2dvector([0,0],[1,2],.1,.04,\n   \+
               thickness=2,color=COLOR(RGB,0,.8,0)):\np2 := plot2dvect
or([0,0],[6,3],.1,.02,\n                       thickness=2,color=brown
):\np3 := plot2dvector([1,2],[6,3],.1,.02,\n                      thic
kness=2,color=red):\nt1 := plots[textplot]([[-.08,-.08,`O`],[.6,2.25,`
P (x  ,y  ,z  )`],\n       [6.1,3.25,`P (x  ,y  ,z  )`]],\n   color=bl
ack,font=[HELVETICA,10]):\nt2 := plots[textplot]([[.6,2.17,`1    1    \+
1    1`],\n       [6.1,3.17,`2     2    2    2`]],\n   color=black,fon
t=[HELVETICA,8]):\nt3 := plots[textplot]([3,2.65,`P P`],color=red,\n  \+
                             font=[HELVETICA,BOLD,10]):\nt4 := plots[t
extplot]([3,2.57,`  1   2`],color=red,\n                              \+
 font=[HELVETICA,8]):\nplots[display]([p1,p2,p3,t1,t2,t3,t4],axes=none
);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 8 "sphere  " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 792 "p1 := plottools[arrow]([1.2,1.3,1.1],\n    [1.2,1.3+1/sqrt(2),1
.1+1/sqrt(2)],.1,.1,.1,\n               arrow,color=red,thickness=1):
\np2 := plottools[arrow]([0,0,-.5],[0,0,2.5],.1,.1,.04,arrow,color=bla
ck):\np3 := plottools[arrow]([0,-.5,0],[0,2.5,0],.1,.1,.04,arrow,color
=black):\np4 := plottools[arrow]([-.5,0,0],[2.5,0,0],.1,.1,.04,arrow,c
olor=black):\np5 := plot3d([1.2+sin(s)*cos(t),1.3+sin(s)*sin(t),1.1+co
s(s)],s=0..Pi,t=0..2*Pi,\n style=patch,grid=[25,25],linestyle=2,color=
COLOR(RGB,.93,.9,.9)):\nt1 := plots[textplot3d]([[2.3,0,-.2,`x`],[0,2.
3,-.2,`y`],\n     [0,-.2,2.2,`z`]],color=black):\nt2 := plots[textplot
3d]([[1.2,2.1,2.3,`centre: (a,b,c)`],\n     [1.2,2.1,2.1,`radius: r`]]
,color=black):\nplots[display]([p1,p2,p3,p4,p5,t1,t2],axes=none,\n    \+
    lightmodel=light4,orientation=[25,70]);" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 0 "" 0 "" {TEXT -1 10 "unit cube " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1325 "p1 := plottools[arrow
]([0,0],[0,4],.1,.15,.07,\n                    arrow,color=red,thickne
ss=2):\np2 := plottools[arrow]([0,0],[-2,-2],.1,.15,.07,\n            \+
        arrow,color=red,thickness=2):\np3 := plottools[arrow]([0,0],[4
,0],.1,.15,.07,\n                    arrow,color=red,thickness=2):\np4
 := plottools[arrow]([0,0],[0,5.3],.1,.13,.05,\n                    ar
row,color=black):\np5 := plottools[arrow]([0,0],[-3,-3],.1,.13,.05,\n \+
                   arrow,color=black):\np6 := plottools[arrow]([0,0],[
5.3,0],.1,.13,.05,\n                    arrow,color=black):\np7 := plo
t([[[-2,-2],[0,4],[4,0],[-2,-2]],\n      [[0,4],[4,4],[2,2],[-2,2],[0,
4]],\n      [[-2,-2],[-2,2]],[[2,2],[2,-2]],\n      [[-2,-2],[2,-2],[4
,0],[4,4]],[[0,4],[1,-1]]],\n      linestyle=2,color=black):\np8 := pl
ot([[0,0],[2/3,2/3]],color=COLOR(RGB,0,.8,0),thickness=2):\np9 := plot
([[2/3,2/3],[2,2]],color=COLOR(RGB,0,.8,0),linestyle=2):\nt1 := plots[
textplot]([[-2.6,-2.9,`x`],[4.9,-.25,`y`],\n       [-.25,4.9,`z`],[-1.
95,-2.2,`A`],[4.1,-.2,`B`],\n         [-.2,4.15,`C`],[1.05,-1.2,`M`],[
-.2,.1,`O`],\n        [.87,.68,`G`],[2.7,2,`P(1,1,1)`]],\n        colo
r=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-.2,1.8,`k`],[1
.75,-.2,`j`],\n        [-.65,-.95,`i`]],color=red,font=[HELVETICA,BOLD
,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,t1,t2],axes=none); \+
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 145012996 145013076 
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