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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Vectors and vector algebra " }}
{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, 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 20 "Scalars and vectors " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 162 "Physical quantitie
s such as length, mass, energy, temperature are completely determined \+
by a numerical magnitude in appropriate units. Such quantities are cal
led " }{TEXT 259 7 "scalars" }{TEXT -1 158 ". Other quantities such as
 force, velocity and acceleration are only completely defined when bot
h a numerical magnitude and a direction in space is specified." }}
{PARA 0 "" 0 "" {TEXT -1 28 "These quantities are called " }{TEXT 259 
7 "vectors" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "It is costu
mary to represent a vector by a " }{TEXT 259 21 "directed line segment
" }{TEXT -1 86 ". Vectors with the same length and direction are consi
dered to be identical. The term " }{TEXT 259 11 "free vector" }{TEXT 
-1 114 " is used in this connection in order to emphasise that the act
ual location of a vector in space is not important. " }}{PARA 256 "" 
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Q\"FF\\uF\\tF]u-%*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" }}}{PARA 
256 "" 0 "" {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 82 "The directe
d line segments or arrows AB, CD, OP and EF represent the same vector.
 " }}{PARA 15 "" 0 "" {TEXT -1 112 "The vector represented by the dire
cted line segment AB, with initial point A and terminal point B is den
oted by " }{TEXT 263 2 "AB" }{TEXT -1 15 " in bold face. " }}{PARA 15 
"" 0 "" {TEXT -1 8 "We have " }{TEXT 264 2 "AB" }{TEXT -1 3 " = " }
{TEXT 265 2 "CD" }{TEXT -1 3 " = " }{TEXT 266 2 "OP" }{TEXT -1 3 " = \+
" }{TEXT 267 2 "EF" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Addition of vectors " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 31 "T
o form the sum of two vectors " }{TEXT 268 1 "u" }{TEXT -1 5 " and " }
{TEXT 269 1 "v" }{TEXT -1 54 ", choose any representative directed lin
e segment for " }{TEXT 272 1 "u" }{TEXT -1 77 " with initial point A a
nd terminal point B. Then choose a representative for " }{TEXT 270 1 "
v" }{TEXT -1 83 " whose initial point coincides with B. If the end poi
nt of this representative for " }{TEXT 271 1 "v" }{TEXT -1 69 " is C, \+
then the directed line segment from A to C represents the sum " }
{TEXT 273 1 "u" }{TEXT -1 3 " + " }{TEXT 274 1 "v" }{TEXT -1 13 ". The
 vector " }{TEXT 275 1 "u" }{TEXT -1 3 " + " }{TEXT 276 1 "v" }{TEXT 
-1 57 " does not depend on the choice of representative arrows. " }}
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}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 13 " Thi
s is the " }{TEXT 259 12 "triangle law" }{TEXT -1 29 " for the additio
n of vectors." }}{PARA 257 "" 0 "" {TEXT -1 10 "          " }}{PARA 
257 "" 0 "" {TEXT -1 74 " By completing the parallelogram ABCD with ap
propriate representatives of " }{TEXT 277 1 "u" }{TEXT -1 5 " and " }
{TEXT 278 1 "v" }{TEXT -1 14 ", we see that " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{TEXT 279 1 "u" }{TEXT -1 3 " + " }{TEXT 280 1 "v" }
{TEXT -1 3 " = " }{TEXT 281 1 "v" }{TEXT -1 3 " + " }{TEXT 282 1 "u" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "The addition of vectors
 is therefore " }{TEXT 259 11 "commutative" }{TEXT -1 2 ". " }}{PARA 
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{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 25 "Give
n representatives of " }{TEXT 299 1 "u" }{TEXT -1 5 " and " }{TEXT 
300 1 "v" }{TEXT -1 80 " with the same initial point, we can obtain a \+
representative vector for the sum " }{TEXT 301 1 "u" }{TEXT -1 3 " + \+
" }{TEXT 302 1 "v" }{TEXT -1 40 " with the same initial point as that \+
of " }{TEXT 303 1 "u" }{TEXT -1 5 " and " }{TEXT 304 1 "v" }{TEXT -1 
57 " by completing a parallelogram like ABCD in the picture. " }}
{PARA 0 "" 0 "" {TEXT -1 48 "This method of adding two vectors is call
ed the " }{TEXT 259 17 "parallelogram law" }{TEXT -1 29 " for the adit
ion of vectors. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 4 "The " }{TEXT 259 11 "associative" }{TEXT -1 5 " law " }}
{PARA 256 "" 0 "" {TEXT -1 3 " ( " }{TEXT 283 1 "u" }{TEXT -1 3 " + " 
}{TEXT 284 1 "v" }{TEXT -1 5 " ) + " }{TEXT 285 1 "w" }{TEXT -1 3 " = \+
" }{TEXT 286 1 "u" }{TEXT -1 5 " + ( " }{TEXT 287 1 "v" }{TEXT -1 3 " \+
+ " }{TEXT 288 1 "w" }{TEXT -1 3 " ) " }}{PARA 257 "" 0 "" {TEXT -1 
46 "also holds as suggested by the picture . . .  " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{GLPLOT2D 371 284 284 {PLOTDATA 2 "64-%'CURVESG6&7$7
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Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "
Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve \+
14" "Curve 15" "Curve 16" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 23 "Subtraction of vectors " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 8 "
negative" }{TEXT -1 13 " of a vector " }{TEXT 289 1 "u" }{TEXT -1 58 "
 is defined to be the vector which has the same length as " }{TEXT 
290 1 "u" }{TEXT -1 45 ", but with the opposite direction to that of \+
" }{TEXT 292 1 "u" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Thi
s vector is denoted by " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"
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{TEXT -1 18 "Given two vectors " }{TEXT 297 1 "u" }{TEXT -1 5 " and " 
}{TEXT 298 1 "v" }{TEXT -1 12 ", we define " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{TEXT 293 1 "v" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F
$!\"\"" }{TEXT 294 1 "u" }{TEXT -1 3 " = " }{TEXT 296 1 "v" }{TEXT -1 
4 " + (" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 295 1 "u
" }{TEXT -1 2 ")." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 364 
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STYLEG6#%,PATCHNOGRIDG-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6&7$F'7$$
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o&*GTF6$!+MP)Qf&F3F?-%&COLORG6&FFF)$\"\")FQF)-F$6&7$F'7$$\"\"(F)$!\"&F
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{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "Note t
hat  " }{TEXT 305 1 "u" }{TEXT -1 5 " + ( " }{TEXT 306 1 "v" }
{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 307 1 "u" }{TEXT 
-1 5 " ) = " }{TEXT 308 1 "v" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Also, " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }
{TEXT -1 2 "( " }{TEXT 312 1 "v" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"
F$!\"\"" }{TEXT 311 1 "u" }{TEXT -1 5 " ) = " }{TEXT 310 1 "u" }
{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 309 1 "v" }{TEXT 
-1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 405 268 268 
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e 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Cur
ve 11" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
95 "A vector can be used to specify the location of a point in a plane
 ( or space ) relative to an " }{TEXT 259 6 "origin" }{TEXT -1 3 " O.
" }}{PARA 0 "" 0 "" {TEXT -1 36 "Vectors used in this way are called \+
" }{TEXT 259 16 "position vectors" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 48 "Thus, in the picture, U has the position vector " }{TEXT 
313 2 "OU" }{TEXT -1 3 " = " }{TEXT 314 1 "u" }{TEXT -1 50 " ( relativ
e to O ), and V has the position vector " }{TEXT 315 2 "OV" }{TEXT -1 
3 " = " }{TEXT 316 1 "v" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
10 "Note that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 317 1 "v" }
{XPPEDIT 18 0 "`` - ``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 318 1 "u" }
{TEXT -1 3 " = " }{TEXT 319 2 "OV" }{XPPEDIT 18 0 "`` - ``" "6#,&%!G\"
\"\"F$!\"\"" }{TEXT 320 2 "OU" }{TEXT -1 3 " = " }{TEXT 321 2 "UV" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{TEXT 322 1 "u" }{XPPEDIT 18 0 "`` - ``" "6#,&%!G
\"\"\"F$!\"\"" }{TEXT 326 1 "v" }{TEXT -1 3 " = " }{TEXT 323 2 "OU" }
{XPPEDIT 18 0 "`` - ``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 324 2 "OV" }
{TEXT -1 3 " = " }{TEXT 325 2 "VU" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 119 "The upper case letters on the right of these equations a
re in the opposite order to the lower case letters on the left." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Multiplication of a vec
tor by a scalar      " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}{PARA 0 "" 0 "" {TEXT -1 13 "For a vector " }{TEXT 329 1 "u" }{TEXT 
-1 15 ", we can write " }{TEXT 327 1 "u" }{TEXT -1 3 " + " }{TEXT 328 
1 "u" }{TEXT -1 5 " = 2 " }{TEXT 330 1 "u" }{TEXT -1 9 ". Thus 2 " }
{TEXT 331 1 "u" }{TEXT -1 47 " is a vector which is in the same direct
ion as " }{TEXT 332 1 "u" }{TEXT -1 31 ", but with twice the length of
 " }{TEXT 333 1 "u" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "Si
milarly, " }{TEXT 336 1 "u" }{TEXT -1 3 " + " }{TEXT 337 1 "u" }{TEXT 
-1 3 " + " }{TEXT 338 1 "u" }{TEXT -1 5 " = 3 " }{TEXT 339 1 "u" }
{TEXT -1 47 " is a vector which is in the same direction as " }{TEXT 
334 1 "u" }{TEXT -1 37 ", but with three times the length of " }{TEXT 
335 1 "u" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "We already k
now that " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 341 1 
"u" }{XPPEDIT 18 0 "``= -1" "6#/%!G,$\"\"\"!\"\"" }{TEXT -1 1 " " }
{TEXT 340 1 "u" }{TEXT -1 24 " has the same length as " }{TEXT 342 1 "
u" }{TEXT -1 36 ", but opposite direction to that of " }{TEXT 343 1 "u
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then (" }{XPPEDIT 18 
0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 344 1 "u" }{TEXT -1 6 " ) + (
" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 345 1 "u" }
{TEXT -1 1 ")" }{XPPEDIT 18 0 "`` = -2" "6#/%!G,$\"\"#!\"\"" }{TEXT 
-1 1 " " }{TEXT 346 1 "u" }{XPPEDIT 18 0 "`` = - ``" "6#/%!G,$F$!\"\"
" }{TEXT -1 4 "( 2 " }{TEXT 347 1 "u" }{TEXT -1 77 " ) is a vector wit
h twice the length of u, but opposite direction to that of " }{TEXT 
348 1 "u" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
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en  " }{TEXT 351 1 "s" }{TEXT -1 1 " " }{TEXT 350 1 "u" }{TEXT -1 37 "
  denotes the vector whose length is " }{XPPEDIT 18 0 "abs(s)" "6#-%$a
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{TEXT -1 44 " and whose direction is the same as that of " }{TEXT 355 
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 " }{TEXT 359 1 "v" }{TEXT -1 23 " are two vectors, then " }}{PARA 
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363 1 "u" }{XPPEDIT 18 0 "`` = t;" "6#/%!G%\"tG" }{TEXT -1 3 " ( " }
{TEXT 365 1 "s" }{TEXT -1 1 " " }{TEXT 364 1 "u" }{TEXT -1 4 " ), " }}
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The last of these equatio
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mples of geometrical proofs involving vector algebra " }}{EXCHG {PARA 
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"Example 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 
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$\")#)eqk!\")$\"))eqk\"F`oFaoFJ-F$6'7$F*7$$\"#5F)$\"\"&F)7%7$$\"+Ux$>v
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$6'7$FPFfo7%7$$\"+`:]R))F6$\"+C#\\-)GF67$$FHF)$\"\"$F)7$$\"+Z%)\\+!*F6
$\"+w2v*z#F6F?FCFJ-F$6%7$F*FP-F\\o6&FFF)F)F)-%*LINESTYLEGFL-F$6%7$F'Ff
oF`rFbr-F$6&7#7$$\"3E++++++!y%!#<$\"3!*************HEF]s-%'SYMBOLG6#%'
CIRCLEGF`r-F@6#%&POINTG-F$6&Fir-Fas6#%(DIAMONDGF`rFds-F$6&Fir-Fas6#%&C
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15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" 
"Curve 22" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 113 "Consider the parallelogram OUTV shown in the picture. Le
t the mid-point of OT be M, and the mid-point of UV be N." }}{PARA 0 "
" 0 "" {TEXT -1 4 "Let " }{TEXT 377 1 "u" }{TEXT -1 54 " be the positi
on vector of U (relative to O), and let " }{TEXT 378 1 "v" }{TEXT -1 
35 " be the position vector of V. Thus " }{TEXT 379 2 "OU" }{TEXT -1 
3 " = " }{TEXT 380 1 "u" }{TEXT -1 5 " and " }{TEXT 381 2 "OV" }{TEXT 
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-1 4 "Now " }{TEXT 383 2 "UV" }{TEXT -1 3 " = " }{TEXT 384 1 "v" }
{XPPEDIT 18 0 " ``-`` " "6#,&%!G\"\"\"F$!\"\"" }{TEXT 385 1 "u" }
{TEXT -1 38 ", and since N is the mid-point of UV, " }{TEXT 386 2 "UN
" }{XPPEDIT 18 0 " ``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 "
 " }{TEXT 402 2 "UV" }{XPPEDIT 18 0 " ``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!
\"\"" }{TEXT -1 3 " ( " }{TEXT 387 1 "v" }{XPPEDIT 18 0 " ``-`` " "6#,
&%!G\"\"\"F$!\"\"" }{TEXT 388 1 "u" }{TEXT -1 3 " )." }}{PARA 0 "" 0 "
" {TEXT -1 6 "Hence " }{TEXT 389 2 "ON" }{TEXT -1 3 " = " }{TEXT 390 
2 "OU" }{TEXT -1 3 " + " }{TEXT 391 2 "UN" }{TEXT -1 3 " = " }{TEXT 
392 1 "u" }{XPPEDIT 18 0 " ``+ 1/2" "6#,&%!G\"\"\"*&F%F%\"\"#!\"\"F%" 
}{TEXT -1 3 " ( " }{TEXT 393 1 "v" }{XPPEDIT 18 0 " ``-`` " "6#,&%!G\"
\"\"F$!\"\"" }{TEXT 394 1 "u" }{TEXT -1 5 " ) = " }{TEXT 395 1 "u" }
{XPPEDIT 18 0 " ``+ 1/2" "6#,&%!G\"\"\"*&F%F%\"\"#!\"\"F%" }{TEXT -1 
1 " " }{TEXT 396 1 "v" }{XPPEDIT 18 0 " ``- 1/2" "6#,&%!G\"\"\"*&F%F%
\"\"#!\"\"F(" }{TEXT -1 1 " " }{TEXT 397 1 "u" }{XPPEDIT 18 0 " ``= 1/
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{XPPEDIT 18 0 " ``+ 1/2" "6#,&%!G\"\"\"*&F%F%\"\"#!\"\"F%" }{TEXT -1 
1 " " }{TEXT 399 1 "v" }{XPPEDIT 18 0 " ``= 1/2" "6#/%!G*&\"\"\"F&\"\"
#!\"\"" }{TEXT -1 3 " ( " }{TEXT 400 1 "u" }{TEXT -1 3 " + " }{TEXT 
401 1 "v" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 19 "On the other hand, " }{TEXT 403 2 "OT" }{TEXT 
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-1 39 ", and, since M is the mid-point of OT, " }{TEXT 406 2 "OM" }
{XPPEDIT 18 0 " ``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " 
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{TEXT 413 2 "ON" }{TEXT -1 71 ". Since M and N have the same position \+
vector, they are the same point." }}{PARA 0 "" 0 "" {TEXT -1 86 "We co
nclude that the diagonals UV and OT of the parallelogram OUTV bisect e
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{TEXT -1 88 "A line joining a vertex of a triangle to the mid-point of
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" }}{PARA 0 "" 0 "" {TEXT -1 168 "We show, using vectors, that the thr
ee medians of a triangle intersect at a single point two-thirds of the
 way along each median from each vertex to the opposite side. " }}
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{PARA 256 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 107 "Cons
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}{TEXT 416 3 "OU " }{TEXT -1 2 "= " }{TEXT 417 2 "u " }{TEXT -1 4 "and
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 = " }{TEXT 431 2 "OM" }{XPPEDIT 18 0 " ``-`` " "6#,&%!G\"\"\"F$!\"\"
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{TEXT -1 5 "Then " }{TEXT 435 2 "UG" }{XPPEDIT 18 0 " ``= 2/3" "6#/%!G
*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 436 2 "UM" }{XPPEDIT 
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1 "u" }{TEXT -1 2 " )" }{XPPEDIT 18 0 "`` = 1/3" "6#/%!G*&\"\"\"F&\"\"
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{TEXT -1 6 ",  so " }{TEXT 441 2 "OG" }{TEXT -1 3 " = " }{TEXT 442 2 "
OU" }{TEXT -1 3 " + " }{TEXT 443 2 "UG" }{TEXT -1 3 " = " }{TEXT 444 
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{TEXT -1 1 " " }{TEXT 445 1 "v" }{XPPEDIT 18 0 " ``- 2/3" "6#,&%!G\"\"
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447 1 "u" }{TEXT -1 3 " + " }{TEXT 448 1 "v" }{TEXT -1 4 " ). " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "On the ot
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}{TEXT 451 1 "v" }{TEXT -1 36 " )   (see the previous example), so " }
{TEXT 452 2 "OH" }{XPPEDIT 18 0 " ``= 2/3" "6#/%!G*&\"\"#\"\"\"\"\"$!
\"\"" }{TEXT -1 1 " " }{TEXT 453 2 "ON" }{XPPEDIT 18 0 "`` = 1/3" "6#/
%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 3 " ( " }{TEXT 454 1 "u" }{TEXT -1 
3 " + " }{TEXT 455 1 "v" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "From this we see that " }{TEXT 
424 2 "OH" }{XPPEDIT 18 0 "`` = 1/3;" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }
{TEXT -1 3 " ( " }{TEXT 425 1 "u" }{TEXT -1 3 " + " }{TEXT 426 1 "v" }
{TEXT -1 5 " ) = " }{TEXT 427 2 "OG" }{TEXT -1 71 ". Since G and H hav
e the same position vector, they are the same point." }}{PARA 0 "" 0 "
" {TEXT -1 146 "We conclude that any two medians of a triangle interse
ct at a point two-thirds of the way along each median from each vertex
 to the opposite side." }}{PARA 0 "" 0 "" {TEXT -1 53 "Hence all three
 medians intersect at a single point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{GLPLOT2D 341 226 226 {PLOTDATA 2 "60-%'CURVESG6'7$7$$
\"\"!F)F(7$$\"\")F)F(7%7$$\"++++gx!\"*$\"+++++&)!#6F*7$F/$!+++++&)F4-%
&STYLEG6#%,PATCHNOGRIDG-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNES
SG6#\"\"#-F$6'7$F'7$$\"\"'F)FK7%7$$\"+Cf*)fdF1$\"+wS5!)eF1FJ7$FQFOF8-F
=6&F?F(F(F@FC-F$6%7$7$$\"\"%F)F(7$$\"\"$F)Fgn-F=6&F?F)F)F)-%*LINESTYLE
GFE-F$6%7$F*FJFinF[o-%%TEXTG6&7$$\"#C!\"\"$\"$&H!\"#Q\"v6\"FT-%%FONTG6
%%*HELVETICAG%%BOLDG\"#5-Fao6&7$$\"$b$Fio$!#DFioQ\"uF[pF<F\\p-Fao6&7$$
FioFfoF(Q\"OF[pFin-F]p6$F_pFap-Fao6&7$$\"#UFfo$\"#NFioQ\"MF[pFinF_q-Fa
o6&7$F+$!\"$FfoQ\"UF[pFinF_q-Fao6&7$$\"#LFfo$\"#HFfoQ\"NF[pFinF_q-Fao6
&7$$\"#fFfo$\"#jFfoQ\"VF[pFinF_q-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$
Q!F[pFfs-%%VIEWG6$%(DEFAULTGFjs" 1 2 0 1 10 0 2 9 1 1 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }}}
{PARA 0 "" 0 "" {TEXT 492 1 "u" }{TEXT -1 5 " and " }{TEXT 493 1 "v" }
{TEXT -1 109 " are the position vectors of U and V respectively, and M
 and N are the mid-points of OU and OV respectively. " }}{PARA 0 "" 0 
"" {TEXT -1 12 "(a) Express " }{TEXT 456 2 "UV" }{TEXT -1 2 ", " }
{TEXT 490 2 "OM" }{TEXT -1 2 ", " }{TEXT 491 2 "ON" }{TEXT -1 5 " and \+
" }{TEXT 489 2 "MN" }{TEXT -1 25 " in terms of the vectors " }{TEXT 
457 1 "u" }{TEXT -1 5 " and " }{TEXT 458 1 "v" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 129 "(b) Prove that the line joining the mid-
points of two sides of a triangle is parallel to the third side and ha
s half its length. " }}{PARA 0 "" 0 "" {TEXT -1 44 "__________________
__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 
"____________________________________________" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{GLPLOT2D 444 271 271 {PLOTDATA 2 "65-%'CURVESG6'7$7$$
\"\"!F)F(7$$\"#7F)F(7%7$$\"++++k6!\")$\"+++++&)!#6F*7$F/$!+++++&)F4-%&
STYLEG6#%,PATCHNOGRIDG-%&COLORG6&%$RGBGF)$\"\")!\"\"F)-%*THICKNESSG6#
\"\"#-F$6'7$F'7$$FFF)$\"\"'F)7%7$$\"+?>O>=!\"*$\"+g$zos&FRFJ7$$\"+!3Q1
)>FR$\"+S17tcFRF8-%'COLOURG6&F?$\"*++++\"F1F(F(FC-F$6'7$FJ7$$FAF)F]o7%
7$$\"+S17twFR$\"+!3Q1)zFRF\\o7$$\"+g$zos(FR$\"+?>O>yFRF8-Fen6&F?F(F(Fg
nFC-F$6%7'7$$\"\"\"F)$\"\"$F)7$$\"\"&F)$\"\"(F)7$$\"#5F)$\"\"%F)7$FLF(
F^p-Fen6&F?F)F)F)-%*LINESTYLEGFE-F$6%7$F\\oF*F^qF`q-%%TEXTG6&7$$\"#&*!
\"#$\"#WFBQ\"u6\"FZ-%%FONTG6%%*HELVETICAG%%BOLDGFjp-Ffq6&7$$\"#QFB$\"#
sFBQ\"vF_rFioF`r-Ffq6&7$$\"#wFB$!\"%FBQ\"wF_rF<F`r-Ffq6&7$$!\"$FB$FBFB
Q\"OF_rF^q-Far6$FcrFjp-Ffq6&7$$\"#=FB$\"#lFBQ\"AF_rF^qF\\t-Ffq6&7$$\"#
#)FB$\"#&)FBQ\"BF_rF^qF\\t-Ffq6&7$$\"$B\"FBFjsQ\"CF_rF^qF\\t-Ffq6&7$$F
MFB$\"#JFBQ\"MF_rF^qF\\t-Ffq6&7$FdpFctQ\"NF_rF^qF\\t-Ffq6&7$$\"$/\"FB$
\"#TFBQ\"PF_rF^qF\\t-Ffq6&7$$\"#fFBFguQ\"QF_rF^qF\\t-%*AXESSTYLEG6#%%N
ONEG-%+AXESLABELSG6$Q!F_rFdw-%%VIEWG6$%(DEFAULTGFhw" 1 2 0 1 10 0 2 9 
1 1 2 1.000000 46.000000 46.000000 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" }}}
{PARA 0 "" 0 "" {TEXT -1 89 "The points M, N, P and Q are the mid-poin
ts of the four sides of the quadrilateral OABC. " }}{PARA 0 "" 0 "" 
{TEXT 467 2 "OA" }{TEXT -1 3 " = " }{TEXT 468 1 "u" }{TEXT -1 2 ", " }
{TEXT 469 2 "AB" }{TEXT -1 3 " = " }{TEXT 470 1 "v" }{TEXT -1 5 " and \+
" }{TEXT 471 2 "OC" }{TEXT -1 3 " = " }{TEXT 472 1 "w" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 46 "(a) Express the following vectors in \+
terms of " }{TEXT 473 1 "u" }{TEXT -1 2 ", " }{TEXT 474 1 "v" }{TEXT 
-1 5 " and " }{TEXT 475 1 "w" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 7 "   (i) " }{TEXT 476 2 "MA" }{TEXT -1 6 " (ii) " }{TEXT 
477 2 "AN" }{TEXT -1 7 " (iii) " }{TEXT 478 2 "MN" }{TEXT -1 6 " (iv) \+
" }{TEXT 479 2 "MQ" }{TEXT -1 5 " (v) " }{TEXT 480 2 "OB" }{TEXT -1 6 
" (vi) " }{TEXT 481 2 "BC" }{TEXT -1 7 " (vii) " }{TEXT 482 2 "BP" }
{TEXT -1 8 " (viii) " }{TEXT 483 2 "NP" }{TEXT -1 6 " (ix) " }{TEXT 
484 2 "QP" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "(b) By comp
aring the vector " }{TEXT 485 2 "MN" }{TEXT -1 6 " with " }{TEXT 486 
2 "QP" }{TEXT -1 16 " and the vector " }{TEXT 487 2 "MQ" }{TEXT -1 6 "
 with " }{TEXT 488 2 "NP" }{TEXT -1 36 " prove that MNPQ is a parallel
ogram." }}{PARA 0 "" 0 "" {TEXT -1 132 "(c) Prove that the sides of th
e parallelogram MNPQ are parallel to, and half the length of the diago
nals of the quadrilateral OABC. " }}{PARA 0 "" 0 "" {TEXT -1 90 "(d) G
ive an alternative proof of the result in (c) by using the conclusion \+
of question 1. " }}{PARA 0 "" 0 "" {TEXT -1 44 "______________________
______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 44 "_____________________________________
_______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 55 "The position vectors of the \+
four points U, V, P, Q are " }{TEXT 459 1 "u" }{TEXT -1 2 ", " }{TEXT 
460 1 "v" }{TEXT -1 5 ",  2 " }{TEXT 461 1 "u" }{TEXT -1 5 " + 3 " }
{TEXT 462 1 "v" }{TEXT -1 2 ", " }{TEXT 463 1 "u" }{XPPEDIT 18 0 " ``-
 2" "6#,&%!G\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 464 1 "v" }{TEXT 
-1 15 " respectively. " }}{PARA 0 "" 0 "" {TEXT -1 50 "Express UP, QV,
 VP and PQ in terms of the vectors " }{TEXT 465 1 "u" }{TEXT -1 5 " an
d " }{TEXT 466 1 "v" }{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 ";
" }}}}{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 pictures" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 260 13 "plo
t2dvector " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 993 "plot2dvector := proc(point::\{Vector, vector, list
\}, vect::\{Vector, vector, list\},head_width,head_relative_height)\n \+
  local a, b, i, x, y, Cos, Sin, v, locopts, 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(point,Vector(2,[a-x,b-y]),\n      \+
  head_width,head_relative_height,args[5..-1])\n   end if;\n   L := ev
alf(sqrt(a^2+b^2));\n   if a=0 and b=0 then return CURVES([]) end if;
\n   Cos := evalf(a/L);\n   Sin := evalf(b/L);\n   v := [[[x,y],[x+a,y
+b]],[[x-1/2*head_width*Sin \n      -head_relative_height*Cos*L+1/2*a,
\n        y+1/2*head_width*Cos-head_relative_height*Sin*L+1/2*b],\n   \+
   [x+1/2*a,y+1/2*b],  \n        [x+1/2*head_width*Sin-head_relative_h
eight*Cos*L+1/2*a, \n        y-1/2*head_width*Cos -head_relative_heigh
t*Sin*L+1/2*b]]];\n    v := evalf(v);\n    locopts := [args[5 .. -1]];
\n    locopts := \n       convert(['style'='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 "equal vectors  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
876 "p1 := plot2dvector([0,0],[3,1],.2,.06,color=red,\n               \+
                    thickness=2):\np2 := plot2dvector([1,2],[4,3],.2,.
06,color=blue,\n                                   thickness=2):\np3 :
= plot2dvector([-1,3],[2,4],.2,.06,color=brown,\n                     \+
               thickness=2):\np4 := plot2dvector([-2,-2],[1,-1],.2,.06
,\n               color=COLOR(RGB,0,.8,0),thickness=2):\np5 := plottoo
ls[arrow]([-3,0],[5,0],0,.15,.02,\n                                arr
ow,color=black):\np6 := plottools[arrow]([0,-3],[0,5],0,.15,.02,\n    \+
                            arrow,color=black):\nt1 := plots[textplot]
([[5,-.2,`x`],[-.2,5,`y`],\n       [-1.2,3,`A`],[2.2,4.1,`B`],[.8,2,`C
`],[4.2,3.1,`D`],\n       [-.2,-.2,`O`],[3.2,1.1,`P`],\n       [-2.2,-
2,`E`],[1.2,-.9,`F`]],\n                    color=black,font=[HELVETIC
A,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,t1],axes=none);" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "addition of vectors  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
875 "p1 := plot2dvector([0,0],[2,4],.23,.05,color=red):\np2 := plot2dv
ector([-2,2],[0,6],.23,.05,color=red):\np3 := plot2dvector([2,4],[11,3
],.2,.04,\n                        color=COLOR(RGB,0,.8,0)):\np4 := pl
ot2dvector([4,6],[13,5],.2,.04,\n                        color=COLOR(R
GB,0,.8,0)):\np5 := plot2dvector([0,0],[11,3],.17,.03,color=blue):\nt1
 := plots[textplot]([[-.18,-.2,`A`],[2,4.3,`B`],[11.3,3,`C`]],\n      \+
              color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]
([[-1.3,4.2,`u`],[.73,2.2,`u`]],\n                    color=red,font=[
HELVETICA,BOLD,10]):\nt3 := plots[textplot]([[6.2,3.3,`v`],[8.3,5.3,`v
`]],\n        color=COLOR(RGB,0,.8,0),font=[HELVETICA,BOLD,10]):\nt4 :
= plots[textplot]([5.7,1.2,`u + v`],\n               color=blue,font=[
HELVETICA,BOLD,10]):\nplots[display]([p1,p2,p3,p4,p5,t1,t2,t3,t4],axes
=none,\n                                   thickness=2);" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 870 "p1 := plot2dvector([0,0],[2,4],.2,.06,colo
r=red):\np2 := plot2dvector([9,-1],[11,3],.2,.06,color=red):\np3 := pl
ot2dvector([2,4],[11,3],.2,.05,\n                        color=COLOR(R
GB,0,.8,0)):\np4 := plot2dvector([0,0],[9,-1],.2,.05,\n               \+
         color=COLOR(RGB,0,.8,0)):\np5 := plot2dvector([0,0],[11,3],.1
7,.04,color=blue):\nt1 := plots[textplot]([[-.18,-.2,`A`],[2,4.3,`B`],
[11.3,3.2,`C`],\n       [9.3,-1.1,`D`]],color=black,font=[HELVETICA,10
]):\nt2 := plots[textplot]([[10.3,1,`u`],[.73,2.2,`u`]],\n            \+
        color=red,font=[HELVETICA,BOLD,10]):\nt3 := plots[textplot]([[
6.2,3.3,`v`],[4.3,-.7,`v`]],\n        color=COLOR(RGB,0,.8,0),font=[HE
LVETICA,BOLD,10]):\nt4 := plots[textplot]([5.7,1.2,`u + v`],\n        \+
       color=blue,font=[HELVETICA,BOLD,10]):\nplots[display]([p1,p2,p3
,p4,p5,t1,t2,t3,t4],axes=none,\n                             thickness
=2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1192 "p1 := plot2dvector([0,
0],[2,4],.3,.05,color=red):\np2 := plot2dvector([2,4],[11,3],.2,.06,\n
                           color=COLOR(RGB,0,.8,0)):\np3 := plot2dvect
or([0,0],[11,3],.18,.045,\n                          color=COLOR(RGB,.
9,0,.9)):\np4 := plot2dvector([11,3],[16,-3],.3,.04,color=blue):\np5 :
= plot2dvector([2,4],[16,-3],.2,.02,color=brown):\np6 := plot2dvector(
[0,0],[16,-3],.2,.03,color=coral):\nt1 := plots[textplot]([[-.18,-.2,`
A`],[2,4.3,`B`],[11.3,3.2,`C`],\n    [16.4,-3,`D`]],color=black,font=[
HELVETICA,10]):\nt2 := plots[textplot]([[.73,2.2,`u`]],\n             \+
       color=red,font=[HELVETICA,BOLD,10]):\nt3 := plots[textplot]([[6
.3,4,`v`]],\n        color=COLOR(RGB,0,.8,0),font=[HELVETICA,BOLD,10])
:\nt4 := plots[textplot]([[13.9,.3,`w`]],\n        color=blue,font=[HE
LVETICA,BOLD,10]):\nt5 := plots[textplot]([4.4,.9,`u + v`],\n         \+
      color=COLOR(RGB,.9,0,.9),font=[HELVETICA,BOLD,10]):\nt6 := plots
[textplot]([10,.7,`v + w`],\n               color=brown,font=[HELVETIC
A,BOLD,10]):\nt7 := plots[textplot]([6.7,-1.8,`u + v + w`],\n         \+
      color=coral,font=[HELVETICA,BOLD,10]):\nplots[display]([p1,p2,p3
,p4,p5,p6,t1,t2,t3,t4,t5,t6,t7],\n                axes=none,thickness=
2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "subtractio
n of vectors  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 447 "p1 := plot2dvector([0,0],[4,2],.1,.03,color=red
):\np2 := plot2dvector([4.2,1.6],[.2,-.4],.1,.03,color=blue):\nt1 := p
lots[textplot]([1.95,1.2,`u`],\n                    color=red,font=[HE
LVETICA,BOLD,10]):\nt2 := plots[textplot]([2.25,.5,`    u`],\n        \+
            color=blue,font=[HELVETICA,BOLD,10]):\nt3 := plots[textplo
t]([2.25,.5,`-`],\n                    color=blue,font=[HELVETICA,13])
:\nplots[display]([p1,p2,t1,t2,t3],axes=none,thickness=2);" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 915 "p
1 := plot2dvector([0,0],[2,4],.2,.06,color=red):\np2 := plot2dvector([
0,0],[9,-1],.2,.04,\n                           color=COLOR(RGB,0,.8,0
)):\np3 := plot2dvector([0,0],[7,-5],.2,.04,\n                        \+
     color=COLOR(RGB,.9,0,.9)):\np4 := plot2dvector([2,4],[9,-1],.2,.0
4,\n                           color=COLOR(RGB,.9,0,.9)):\np5 := plot2
dvector([9,-1],[7,-5],.2,.06,color=blue):\nt1 := plots[textplot]([.85,
2.3,`u`],\n                    color=red,font=[HELVETICA,BOLD,10]):\nt
2 := plots[textplot]([8.1,-3.3,`- u`],\n                    color=blue
,font=[HELVETICA,BOLD,10]):\nt3 := plots[textplot]([4.5,-.1,`v`],\n   \+
           color=COLOR(RGB,0,.8,0),font=[HELVETICA,BOLD,10]):\nt4 := p
lots[textplot]([[5.5,2.1,`v - u`],[3.5,-3,`v - u`]],\n              co
lor=COLOR(RGB,.9,0,.9),font=[HELVETICA,BOLD,10]):\nplots[display]([p1,
p2,p3,p4,p5,t1,t2,t3,t4],axes=none,\n                                 \+
        thickness=2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 978 "p1 := plot2dvector([0,0],[2,4],.17
,.04,color=red):\np2 := plot2dvector([0,0],[9,-1],.15,.03,\n          \+
                         color=COLOR(RGB,0,.8,0)):\np3 := plot2dvector
([1.98,3.96],[8.98,-1.04],.2,.025,\n                                  \+
color=COLOR(RGB,.9,0,.9)):\np4 := plot2dvector([9.02,-.96],[2.02,4.04]
,.2,.025,\n                                color=brown):\nt1 := plots[
textplot]([.85,2.2,`u`],\n                    color=red,font=[HELVETIC
A,BOLD,10]):\nt2 := plots[textplot]([4.3,-.7,`v`],\n              colo
r=COLOR(RGB,0,.8,0),font=[HELVETICA,BOLD,10]):\nt3 := plots[textplot](
[4.4,1.8,`v - u`],\n              color=COLOR(RGB,.9,0,.9),font=[HELVE
TICA,BOLD,10]):\nt4 := plots[textplot]([5.9,1.8,`u - v`],\n           \+
   color=brown,font=[HELVETICA,BOLD,10]):\nt5 := plots[textplot]([[-.2
,-.1,`O`],[2,4.35,`U`],[9.3,-.9,`V`]],\n              color=black,font
=[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,t1,t2,t3,t4,t5],axes=no
ne,\n                                  thickness=2);" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 40 "multiplication of a vector by a s
calar  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 776 "p1 := plot2dvector([0,0],[3,1],.1,.05,color=red):\np
2 := plot2dvector([.2,-.4],[6.2,1.6],.1,.03,color=brown):\np3 := plot2
dvector([6.4,1.2],[.4,-.8],.1,.03,\n                               col
or=COLOR(RGB,.9,0,.9)):\nt1 := plots[textplot]([1.3,.7,`u`],\n        \+
            color=red,font=[HELVETICA,BOLD,10]):\nt2 := plots[textplot
]([3.75,1,`u`],\n                    color=brown,font=[HELVETICA,BOLD,
10]):\nt3 := plots[textplot]([3.6,1,`2`],\n                    color=b
rown,font=[HELVETICA,10]):\nt4 := plots[textplot]([3.4,-.1,`u`],\n    \+
     color=COLOR(RGB,.9,0,.9),font=[HELVETICA,BOLD,10]):\nt5 := plots[
textplot]([3.2,-.1,`-2`],\n         color=COLOR(RGB,.9,0,.9),font=[HEL
VETICA,10]):\nplots[display]([p1,p2,p3,t1,t2,t3,t4,t5],axes=none,\n   \+
                          thickness=2);" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1633 "p1 := plottools[arr
ow]([0,0],[4,4],0,.2,.05,\n                            arrow,color=red
):\np2 := plottools[arrow]([-.03,.06],[6.97,7.06],0,.2,.03,\n         \+
                    arrow,color=brown):\np3 := plottools[arrow]([0,0],
[8,-1],0,.2,.04,\n                           arrow,color=COLOR(RGB,0,.
8,0)):\np4 := plottools[arrow]([-.03,-.06],[13.97,-1.81],0,.2,.03,\n  \+
                     arrow,color=COLOR(RGB,.5,0,.9)):\np5 := plottools
[arrow]([-.03,.06],[11.97,3.06],0,.2,.03,\n                        arr
ow,color=blue):\np6 := plottools[arrow]([0,0],[21,5.25],0,.2,.02,\n   \+
          arrow,color=COLOR(RGB,.85,0,.85)):\np7 := plot([[[6.97,7.06]
,[21,5.25],[13.97,-1.81]],\n           [[4,4],[11.97,3.06],[8,-1]]],\n
              color=black,linestyle=2):\nt1 := plots[textplot]([2.34,1
.87,`u`],\n               color=red,font=[HELVETICA,BOLD,10]):\nt2 := \+
plots[textplot]([4.5,-.2,`v`],\n     color=COLOR(RGB,0,.8,0),font=[HEL
VETICA,BOLD,10]):\nt3 := plots[textplot]([4.1,4.9,`u`],\n   color=brow
n,font=[HELVETICA,BOLD,10]):\nt4 := plots[textplot]([3.7,4.9,`s`],\n  \+
 color=brown,font=[HELVETICA,10]):\nt5 := plots[textplot]([9.3,-1.5,`v
`],\n   color=COLOR(RGB,.5,0,.9),font=[HELVETICA,BOLD,10]):\nt6 := plo
ts[textplot]([8.9,-1.5,`s`],\n   color=COLOR(RGB,.5,0,.9),font=[HELVET
ICA,10]):\nt7 := plots[textplot]([6.4,2.2,`u + v`],\n   color=blue,fon
t=[HELVETICA,BOLD,10]):\nt8 := plots[textplot]([14.4,4.25,`u + v`],\n \+
  color=COLOR(RGB,.85,0,.85),font=[HELVETICA,BOLD,10]):\nt9 := plots[t
extplot]([14.2,4.25,`s (         )`],\n   color=COLOR(RGB,.85,0,.85),f
ont=[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,\n        t
1,t2,t3,t4,t5,t6,t7,t8,t9],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "
" 0 "" {TEXT -1 52 "the diagonals of a parallelogram bisect each other
  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1071 "p1 := plot2dvector([0,0],[2,4],.18,.04,\n          \+
     color=COLOR(RGB,.9,0,.9),thickness=2):\np2 := plot2dvector([0,0],
[8,1],.13,.03,\n                          color=brown,thickness=2):\np
3 := plot2dvector([2,4],[10,5],.13,.03,\n                          col
or=brown,thickness=2):\np4 := plot2dvector([8,1],[10,5],.18,.04,\n    \+
            color=COLOR(RGB,.9,0,.9),thickness=2):\np5 := plot([[[2,4]
,[8,1]],[[0,0],[10,5]]],color=black,linestyle=2):\np6 := plot([[[4.78,
2.63]]$3],style=point,\n                   symbol=[circle,diamond,cros
s],color=black):\np7 := plot([[[4.81,2.41]]$3],style=point,\n         \+
          symbol=[circle,diamond,cross],color=black):\nt1 := plots[tex
tplot]([[-.13,-.13,`O`],[2,4.2,`U`],[10.1,5.16,`T`],\n       [8.3,.9,`
V`],[4.8,2.9,`N`],[4.8,2.2,`M`]],color=black,font=[HELVETICA,10]):\nt2
 := plots[textplot]([[.73,2.2,`u`],[9.3,3,`u`]],\n        color=COLOR(
RGB,.85,0,.85),font=[HELVETICA,BOLD,10]):\nt3 := plots[textplot]([[4,.
3,`v`],[5.8,4.75,`v`]],\n      color=brown,font=[HELVETICA,BOLD,10]):
\nplots[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3],axes=none);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 55 "the medians of a triang
le intersect at a single point  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 924 "p1 := plottools[arrow]([0,0
],[4,4],.1,.16,.04,\n             arrow,color=COLOR(RGB,.85,0,.85),thi
ckness=2):\np2 := plottools[arrow]([0,0],[6,-2],.1,.18,.03,\n         \+
               arrow,color=brown,thickness=2):\np3 := plot([[[4,4],[6,
-2]],[[0,0],[5,1]],\n      [[4,4],[3,-1]]],color=black,linestyle=[1,2,
2]):\np4 := plot([[[3.3,.54]]$3],style=point,\n                   symb
ol=[circle,diamond,cross],color=black):\np5 := plot([[[3.48,.7]]$3],st
yle=point,\n                   symbol=[circle,diamond,cross],color=bla
ck):\nt1 := plots[textplot]([[-.13,-.13,`O`],[4,4.2,`U`],[6.2,-2,`V`],
\n       [5.2,1.1,`N`],[3,-1.2,`M`],[3.43,.34,`G`],\n        [3.63,.95
,`H`]],color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([2,2.3
,`u`],\n         color=COLOR(RGB,.85,0,.85),font=[HELVETICA,BOLD,10]):
\nt3 := plots[textplot]([2.4,-1,`v`],\n             color=brown,font=[
HELVETICA,BOLD,10]):\nplots[display]([p1,p2,p3,p4,p5,t1,t2,t3],axes=no
ne);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "triangle p
roblem  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 635 "p1 := plottools[arrow]([0,0],[8,0],.1,.17,.03,\n    \+
           arrow,color=red,thickness=2):\np2 := plottools[arrow]([0,0]
,[6,6],.1,.17,.03,\n               arrow,color=blue,thickness=2):\np3 \+
:= plot([[[4,0],[3,3]],[[8,0],[6,6]]],\n                       color=b
lack,linestyle=2):\nt1 := plots[textplot]([2.4,2.95,`v`],\n           \+
      color=blue,font=[HELVETICA,BOLD,10]):\nt2 := plots[textplot]([3.
55,-.25,`u`],\n                 color=red,font=[HELVETICA,BOLD,10]):\n
t3 := plots[textplot]([[-.2,0,`O`],[4.2,.35,`M`],[8,-.3,`U`],\n  [3.3,
2.9,`N`],[5.9,6.3,`V`]],color=black,font=[HELVETICA,10]):\nplots[displ
ay]([p1,p2,p3,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 
0 "" {TEXT -1 23 "quadrilateral problem  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 889 "p1 := plottools[arro
w]([0,0],[12,0],.1,.17,.03,\n          arrow,color=COLOR(RGB,0,.8,0),t
hickness=2):\np2 := plottools[arrow]([0,0],[2,6],.1,.17,.05,\n        \+
       arrow,color=red,thickness=2):\np3 := plottools[arrow]([2,6],[8,
8],.1,.17,.05,\n          arrow,color=blue,thickness=2):\np4 := plot([
[[1,3],[5,7],[10,4],[6,0],[1,3]],\n    [[8,8],[12,0]]],color=black,lin
estyle=2):\nt1 := plots[textplot]([.95,4.4,`u`],color=red,\n          \+
   font=[HELVETICA,BOLD,10]):\nt2 := plots[textplot]([3.8,7.2,`v`],col
or=blue,\n                   font=[HELVETICA,BOLD,10]):\nt3 := plots[t
extplot]([7.6,-.4,`w`],color=COLOR(RGB,0,.8,0),\n                   fo
nt=[HELVETICA,BOLD,10]):\nt4 := plots[textplot]([[-.3,-.1,`O`],[1.8,6.
5,`A`],[8.2,8.5,`B`],\n  [12.3,-.1,`C`],[.6,3.1,`M`],[5,6.5,`N`],[10.4
,4.1,`P`],\n   [5.9,.6,`Q`]],color=black,font=[HELVETICA,10]):\nplots[
display]([p1,p2,p3,p4,t1,t2,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{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 }