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0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Autho r" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT 256 32 " LINEAR TRANSFORMATIONS OF SPAC E" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 22 "Li near Algebra Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 10 "Objective:" }}{PARA 0 "" 0 "" {TEXT -1 123 "To understand three-dimensional linear transformations; \+ to become familiar with various geometric transformations in space." } }{PARA 0 "" 0 "" {TEXT -1 124 "NOTE: Please make sure you have careful ly worked through the lab on transformations of the plane before worki ng on this lab." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 276 23 "Background Information:" }}{PARA 0 "" 0 "" {TEXT -1 57 "When a 3 \+ x 3 matrix A multiplies a 3 x 1 column vector " }{TEXT 277 1 "v" } {TEXT -1 44 ", the result is another 3 x 1 column vector " }{TEXT 278 1 "w" }{TEXT -1 25 ". We say that the vector " }{TEXT 279 1 "v" } {TEXT -1 10 " has been " }{TEXT 280 11 "transformed" }{TEXT -1 17 " in to the vector " }{TEXT 282 1 "w" }{TEXT -1 12 ", that is, A" }{TEXT 283 1 "v" }{TEXT -1 5 " = " }{TEXT 284 1 "w" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "If we im agine that A multiplies all vectors in space, we get a transformation \+ of all of space. Certain matrices will do predictable things to every \+ vector. For example, if A is a diagonal matrix with all three diagonal entries equal to" }{TEXT 338 2 " r" }{TEXT -1 56 ", it will dilate (s tretch) every vector by a factor of " }{TEXT 339 1 "r" }{TEXT -1 6 ", if " }{TEXT 340 2 "r " }{TEXT -1 13 ">1 . If 0 < " }{TEXT 341 1 "r " }{TEXT -1 81 " <1, every vector will \"shrink.\" In both cases, the \+ direction of the vector will " }{TEXT 285 3 "not" }{TEXT -1 27 " chang e. (What happens if " }{TEXT 342 1 "r" }{TEXT -1 11 " = 1? If " } {TEXT 343 1 "r" }{TEXT -1 10 " = 0? If " }{TEXT 344 2 " r" }{TEXT -1 146 " < 0?) [Note: a diagonal matrix is one in which the diagonal entr ies from upper left to lower right are non-zero, and all other entries are zero.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "A very important theorem you will learn states the following:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 10 "The fi rst " }{TEXT 303 6 "column" }{TEXT 304 108 " of a transformation matri x will be whatever that transformation does to the vector <1, 0, 0>. \+ The second " }{TEXT 305 6 "column" }{TEXT 306 72 " of a transformatio n matrix will be whatever that transformation does to" }{TEXT -1 1 " \+ " }{TEXT 287 22 "the vector <0, 1, 0>." }{TEXT -1 1 " " }{TEXT 323 10 "The third " }{TEXT 325 6 "column" }{TEXT 326 98 " of a transforma tion matrix will be whatever that transformation does to the vector < 0, 0, 1>. " }{TEXT -1 170 "(Does this seem plausible for the dilation matrix discussed above?) This theorem will often make it very easy t o determine the matrix of a transformation when we have a " }{TEXT 324 1 " " }{TEXT 309 21 "geometric description" }{TEXT -1 33 " of what the transformation does." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 281 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 259 15 "Solve d Example:" }}{PARA 0 "" 0 "" {TEXT -1 138 "a) Use the theorem above t o find the matrix of the transformation that rotates every vector coun terclockwise 30 degrees around the z-axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "b) Show a plot of the triangle connecting the points (5, 1, 1), (1, 5, 1), and (9, 9, 9) before and after the above rotation is applied" }{MPLTEXT 0 21 0 "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" } {TEXT 261 9 "Solution:" }{TEXT 262 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 288 0 "" }{TEXT 289 68 "a) When a vector is rota ted around the z-axis, its z-component will " }{TEXT 327 3 "not" } {TEXT 328 204 " change. Its x and y components will behave the same wa y a two dimensional vector would behave when rotated counterclockwise \+ around the origin.Recall that for a two-dimensional vector the compone nts are <" }{TEXT 345 1 "r" }{TEXT 346 5 " cos " }{TEXT 329 0 "" } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 290 2 ", " }{TEXT 291 1 "r " }{TEXT 347 5 " sin " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 1 ">" }{TEXT 292 0 "" }{TEXT 293 8 ", where " }{TEXT 348 2 " r" } {TEXT 349 38 " is the magnitude of the vector and " }{XPPEDIT 18 0 " alpha;" "6#%&alphaG" }{TEXT -1 2 " " }{TEXT 294 0 "" }{TEXT 295 44 "i s the angle that the vector makes with the " }{TEXT 310 8 "positive" } {TEXT 311 63 " x-axis. Using the theorem, <1, 0, 0> will be transform ed into" }}{PARA 4 "" 0 "" {TEXT 297 1 "<" }{XPPEDIT 18 0 "sqrt(3);" " 6#-%%sqrtG6#\"\"$" }{TEXT -1 2 "/ " }{TEXT 308 1 "2" }{TEXT 350 1 "," }{TEXT -1 1 " " }{TEXT 307 6 "1/2, 0" }{TEXT -1 3 ">; " }{TEXT 298 0 " " }{TEXT 299 39 " <0, 1, 0> will be transformed into <" }{TEXT 301 0 "" }{TEXT 300 7 "-1/2, " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\" \"$" }{TEXT -1 2 "/ " }{TEXT 331 0 "" }{TEXT 332 1 "2" }{TEXT -1 3 ", \+ " }{TEXT 333 1 "0" }{TEXT -1 1 ">" }{TEXT 330 2 "; " }{TEXT 334 36 "a nd <0, 0, 1> will stay <0, 0, 1>." }}{PARA 4 "" 0 "" {TEXT 302 49 " \+ So, the matrix of this transformation will be " }{XPPEDIT 18 0 "MATR IX([[sqrt(3)/2, -1/2, 0], [1/2, sqrt(3)/2, 0], [0, 0, 1]]);" "6#-%'MAT RIXG6#7%7%*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\",$*&F-F-F.F/F/\"\"!7%*&F-F- F.F/*&-F*6#F,F-F.F/F27%F2F2F-" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 275 1 " " }{TEXT 296 68 " b) We activate the Maple linear algebra, plots and lamp* packages: " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "with (linal g): with (plots): with (Lamp);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7_ p%'Axis3dG%*BackidmatG%*BacksolveG%(BandmatG%*BasisgridG%$BugG%&Bug3dG %'Bug3dhG%)CharpolyG%+Chevron3dhG%&ClockG%*ColorlistG%.ComponentplotG% )CopyintoG%*CrossprodG%'Cube3dG%(Cube3dhG%$DetG%(DiagmatG%(DotprodG%*D rawlinesG%+DrawmatrixG%+DrawplanesG%(DrawvecG%*Drawvec3dG%&EvalmG%(Eva luesG%)EvectorsG%'ExpandG%*GenmatrixG%)GetcolorG%%GridG%)GridgameG%*Gr idvectsG%)HeadtailG%(Hotel3dG%)Hotel3dhG%&HouseG%)House3dhG%&IdmatG%(I nverseG%&Jet3dG%'Jet3dhG%*JordanmatG%(Lamp3dhG%&LamphG%(LetterLG%(Lett erNG%(LetterSG%(LetterXG%$MagG%)MapcolorG%,MappicturesG%/Matrixtovecto rG%)MatsolveG%&MovieG%*NullbasisG%%PathG%(ProjectG%+ProjectmatG%-Proje ctmat3dG%)QuestionG%(RandmatG%'ReduceG%+ReflectmatG%-Reflectmat3dG%*Re sidualsG%*RotatematG%,Rotatemat3dG%&RowopG%+SymbollistG%+TrajectoryG%- Trajectory3dG%*TransformG%-TranslatematG%/Translatemat3dG%)UnitspanG%* VandermatG%+VectorgridG%+VectorlineG%-VectranslateG%*XshearmatG%*Yshea rmatG" }}}{PARA 0 "" 0 "" {TEXT -1 220 "[Note: if you highlight any of the \"lamp\" commands above and go to the help screen, you will find \+ an explanation of what the command does, the proper format for enterin g it, and executable examples of its use. Try this!]" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 266 "We will use \"m atrix\" ( a regular Maple command) and \"Drawmatrix3d\" and \"Transfor m\" (\"lamp\" commands) to show how we can rotate the given triangle 3 0 degrees counterclockwise around the z-axis. Be sure to click on the \+ pictures and turn them until you get a good view." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "M := matrix( 3,4,[5,1,9,5,1,5,9,1,1,1,9,1]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"MG-%'matrixG6#7%7&\"\"&\"\"\"\"\"*F*7&F+F*F,F+7&F+F+F,F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Drawmatrix(M);\n" }}{PARA 13 "" 1 "" {GLPLOT3D 361 328 328 {PLOTDATA 3 "6)-%)POLYGONSG6&7&7%$\" \"&\"\"!$\"\"\"F*F+7%F+F(F+7%$\"\"*F*F/F/F'-%'COLOURG6&%$RGBG$\"*++++ \"!\")$F*F*F8-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-%'CURVESG6$7$F'F -F1-FB6$7$F-F.F1-FB6$7$F.F'F1-F$6%7#7%F8F8F8-F26&F4F,F,F,F=-%(SCALINGG 6#%,CONSTRAINEDG-%*AXESSTYLEG6#%'NORMALG" 1 2 0 1 10 0 2 1 1 4 1 1.000000 82.000000 50.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "N := matrix ([[sqrt(3)/2, -1/2,0], [1/2,sqrt(3)/2,0],[0,0,1]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'matrixG6#7%7%,$*&\"\"#!\"\"\"\"$#\"\"\"F, F0#F-F,\"\"!7%F/F*F27%F2F2F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Transform(N,M);\n" }}{PARA 13 "" 1 "" {GLPLOT3D 359 260 260 {PLOTDATA 3 "6.-%)POLYGONSG6'7&7%$\"\"&\"\"!$\"\"\"F*F+7%F+F(F+7%$\"\" *F*F/F/F'-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F8-%'SYMBOLG6#%'CIRCLEG- %&STYLEG6#%&POINTG-%*THICKNESSG6#F,-%'CURVESG6%7$F'F-F1FA-FE6%7$F-F.F1 FA-FE6%7$F.F'F1FA-F$6&7#7%F8F8F8-F26&F4F,F,F,F=FA-F$6'7&7%$\"+?q7IQ!\" *$\"+/a-mLFZF+7%$!+'fuRj\"FZ$\"+?q7I[FZF+7%$\"+O'GUH$FZ$\"+kGUH7F7F/FW -F26&F4F8F8F5F9F=-FB6#\"\"$-FE6%7$FWFgnFaoFco-FE6%7$FgnF\\oFaoFco-FE6% 7$F\\oFWFaoFco-F$6&FPFRF=Fco-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6# %'NORMALG" 1 2 0 1 10 0 2 1 1 4 1 1.000000 84.000000 55.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu rve 8" "Curve 9" "Curve 10" }}}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "* Written permission for the use \+ of Linear Algebra Modules Project (LAMP) package from the authors, Her man, et. al. and publisher Addison-Wesley on file." }}}}{PARA 0 "" 0 " " {TEXT -1 77 "_______________________________________________________ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }{TEXT 264 10 "ASSIGNMENT" }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 10 "Problem 1:" } {TEXT 266 0 "" }{TEXT 267 0 "" }{TEXT 268 0 "" }{TEXT 269 0 "" }{TEXT 270 0 "" }{TEXT 271 0 "" }{TEXT 272 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "a) Use the same technique as in the \+ solved example to find the matrix of the transformation that " }{TEXT 312 8 "reflects" }{TEXT -1 62 " every vector across the xy-plane. To \+ reflect a vector across" }}{PARA 0 "" 0 "" {TEXT -1 166 " a plane m eans the following: From the head of the vector draw a perpendicular t o the plane, creating a line segment. Continue this segment a distance equal to its" }}{PARA 0 "" 0 "" {TEXT -1 161 " own length. The poi nt where you are now is the head of the reflected vector. Remember, fi nd what would happen to <1, 0, 0>, <0, 1, 0>, and <0, 0, 1> under" }}{PARA 0 "" 0 "" {TEXT -1 26 " this transformation. " }}{PARA 0 " " 0 "" {TEXT -1 48 " " }}{PARA 0 "" 0 "" {TEXT -1 86 "b) Show a plot of the triangle of the s olved problem before and after the reflection. " }}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 0 "" }{TEXT 273 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 97 "a) Use the same technique as in the solved example to fin d the matrix of the transformation that " }{TEXT 313 8 "projects" } {TEXT -1 14 " every vector " }{TEXT 314 4 "onto" }{TEXT -1 42 " the x z-plane. To project a vector onto a" }}{PARA 0 "" 0 "" {TEXT -1 162 " \+ plane means the following: From the head of the vector draw a perpe ndicular to the plane. Stop there. The point where you are now is the \+ head of the projected" }}{PARA 0 "" 0 "" {TEXT -1 115 " vector. Rem ember, find what would happen to <1, 0, 0>, <0, 1, 0>, and <0, 0, 1> under this transformation. " }}{PARA 0 "" 0 "" {TEXT -1 48 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 219 "b) Show a plot of the triangle of the solved problem before and a fter the projection. \+ \+ " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 274 20 "Problem 3 (Summary):" }{TEXT -1 0 "" }{TEXT 336 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "(A real challenge!) Generalize what you did in part a) of each problem above, including the solved example. That is, find " }{TEXT 315 5 "three" }{TEXT -1 66 " GENERAL matrices: one that rota tes every vector counterclockwise " }{TEXT 316 0 "" }{TEXT 317 5 "thet a" }{TEXT 318 0 "" }{TEXT -1 66 " radians around the z-axis; one that \+ reflects every vector across " }{TEXT 335 3 "any" }{TEXT -1 34 " plan e passing through the origin" }{TEXT 319 0 "" }{TEXT 320 0 "" }{TEXT -1 42 "; and one that projects every vector onto " }{TEXT 337 3 "any" }{TEXT -1 34 " plane passing through the origin" }{TEXT 321 0 "" } {TEXT 322 0 "" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 77 "______ ______________________________________________________________________ _" }}{PARA 0 "" 0 "" {TEXT -1 96 "MSIP Grant #P120A80089-98: \"Three U rban Calculus Reform Programs: Adopting the Best\" 1998-2001" }}} {MARK "7 2 1 0 0" 42 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }