{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 294 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 324 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 325 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 326 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 327 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 328 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 329 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 330 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 331 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 332 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 333 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 334 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 335 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 336 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 337 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 338 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 339 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 340 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 341 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 342 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 343 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 344 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 345 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 346 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 347 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 348 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 349 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 350 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 351 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 352 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 353 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 354 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 355 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 356 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 357 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 358 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 359 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 360 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 361 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 362 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 363 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 364 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 365 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 366 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 367 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 368 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 369 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 370 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 371 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 372 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 373 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 374 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 375 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 376 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 377 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 378 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 379 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 380 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 381 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 382 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 383 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 384 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 385 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 386 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 387 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 388 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 389 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 390 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 391 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 392 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 393 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 394 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 395 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 396 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 397 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 398 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 399 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 400 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 401 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 402 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 403 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 404 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 405 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 406 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "" 256 257 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 259 0 "" }{TEXT 256 35 "Vectors in Two and Three Dimensions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 37 "1. Arithmetic of Vec tors in the Plane" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "First, load t he plots package. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with( plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "1. We show here how vectors are defined and plotted in \+ " }{TEXT 260 5 "Maple" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Enter these two definitions. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "u:=[10,10];v:=[-3,4];" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "No te that " }{TEXT 288 5 "Maple" }{TEXT -1 104 " uses brackets around th e components of the vector, where parentheses are used in conventional notation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "Each vecto r can be plotted as a parametric curve -- actually a straight line. Th e individul components of the vector are \"selected\" by using " } {TEXT 289 5 "Maple" }{TEXT -1 23 "'s subscript notation, " }{TEXT 290 7 "another" }{TEXT -1 24 " use of brackets, as in " }{TEXT 291 0 "" } {TEXT 292 4 "u[1]" }{TEXT -1 9 ". (In a " }{TEXT 293 5 "Maple" } {TEXT -1 49 " command line, this notation would be printed as " } {XPPEDIT 294 0 "u[1];" "6#&%\"uG6#\"\"\"" }{TEXT -1 2 ".)" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 273 21 "\"scaling=constrained\"" }{TEXT -1 172 " option in the following c ommands ensures that the plot will have the true aspect ratio. (This i s the same as clicking on the 1:1 button after the plot has been displ ayed.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "uplot:=plot([t*u[1],t*u[2],t=0..1],thickness=2, colo r=green, scaling=constrained):\nvplot:=plot([t*v[1],t*v[2],t=0..1],thi ckness=2, color=red, scaling=constrained):\ndisplay(uplot,vplot);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Answer the remaining questions in Part 1 here. Use any \+ " }{TEXT 274 5 "Maple" }{TEXT -1 125 " calculations that you find appr opriate -- you may enter them on new command lines below or insert com mand lines as you wish." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "6. (a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "6. (b) " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "6. (c) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "6. (d) " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "7. (mostly on pape r, but there is a question to answer here) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "[Step 8 does not require a resp onse.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "9. (mostly on paper, but there is a question to answer here) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "[Step 10 \+ does not require a response.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "11. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "12. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "13. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 271 18 "2. The Dot Product" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "1. Record your obs ervations from the applet here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 275 5 "Maple" }{TEXT -1 72 " has \+ a dot product function that is part of its linear algebra package, " } {TEXT 276 6 "linalg" }{TEXT -1 96 ". The next block of commands will \+ load the package and compute the dot product of the vectors " }{TEXT 277 1 "u" }{TEXT -1 7 " and " }{TEXT 278 1 "v" }{TEXT -1 32 " as th ey are currently defined." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u; v; dotprod(u,v );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 49 "Answer the remaining steps in Part 2 here, using \+ " }{TEXT 279 5 "Maple" }{TEXT -1 61 " as convenient for either numeric al or symbolic calculations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "4. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "3. Vector-valued Functions and their De rivatives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "We " }{TEXT 368 7 "restart" }{TEXT -1 44 " to clear all variable na mes and reload the " }{TEXT 406 5 "plots" }{TEXT -1 9 " package." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "To define and work with a vector-valued function in Maple, it is conveni ent to work directly with the component functions " }{XPPEDIT 18 0 "x( t);" "6#-%\"xG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y(t);" "6#- %\"yG6#%\"tG" }{TEXT -1 42 " as a parametric description of the curve. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x:=t->t; y:=t->t^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r:=t ->[x(t),y(t)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "We can pl ot this function with the parametric version of the plot command." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot([r(t)[1],r(t)[2],t=-3..3], scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 102 "Note that we addressed the first compon ent of our vector-valued function using the subscript notation " } {TEXT 367 7 "r(t)[1]" }{TEXT -1 43 ", and similarly for the second com ponent. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Answer the first three questions here. If you need to, change the function definition and re-plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 "1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 "2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 "3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "Differentiation can be done directly to t he coordinate functions, using the " }{TEXT 398 1 "D" }{TEXT -1 35 " o perator and the function names. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Dx:=t->D(x)(t); Dy:=t->D(y)( t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Dr:=t->[Dx(t),Dy(t)];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "We have written " }{TEXT 370 2 "Dx" }{TEXT -1 2 ", " }{TEXT 399 2 "Dy" }{TEXT -1 6 ", and " }{TEXT 400 2 "Dr" }{TEXT -1 24 " for the derivatives of " }{TEXT 401 1 "x" } {TEXT -1 2 ", " }{TEXT 402 1 "y" }{TEXT -1 6 ", and " }{TEXT 371 1 "r " }{TEXT -1 30 ", respectively, but these are " }{TEXT 369 10 "just na mes" }{TEXT -1 117 " -- they could be any names. We can use these nam es to calculate position and tangent vectors at any given value of " } {TEXT 403 1 "t" }{TEXT -1 15 ", for example, " }{XPPEDIT 18 0 "t = 2; " "6#/%\"tG\"\"#" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "r(2); Dr(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 74 "What you see in the preceding input-outp ut lines depends on what function " }{TEXT 373 1 "r" }{TEXT -1 33 " cu rrently represents. Redefine " }{TEXT 374 1 "r" }{TEXT -1 54 " and it s component functions above, and also evaluate " }{TEXT 375 1 "r" } {TEXT -1 5 " and " }{TEXT 376 2 "Dr" }{TEXT -1 14 " at values of " } {TEXT 372 1 "t" }{TEXT -1 12 " other than " }{XPPEDIT 18 0 "t = 2;" "6 #/%\"tG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "We can construct and plot a velocity vector by using the components of the derivative vector. In the following calc ulation, we let " }{TEXT 377 0 "" }{TEXT -1 0 "" }{TEXT 378 1 "u" } {TEXT -1 24 " be the position vector " }{TEXT 379 1 "r" }{TEXT -1 42 " (2) to a particular point on the graph of " }{TEXT 380 0 "" }{TEXT 381 1 "r" }{TEXT 382 0 "" }{TEXT -1 13 ", and we let " }{TEXT 383 0 " " }{XPPEDIT 18 0 "v = Dr(2);" "6#/%\"vG-%#DrG6#\"\"#" }{TEXT -1 79 " b e the derivative vector (velocity) at that point. Then we plot the ve locity " }{TEXT 384 1 "v" }{TEXT -1 19 " parametrically as " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "x(s) = u[1]+v[1]*s; " "6#/-%\"xG6#%\"sG,&&%\"uG6#\"\"\"F,*&&%\"vG6#F,F,F'F,F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "y(s) \+ = u[2]+v[2]*s;" "6#/-%\"yG6#%\"sG,&&%\"uG6#\"\"#\"\"\"*&&%\"vG6#F,F-F' F-F-" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 12 " 0 " } {TEXT 385 1 "<" }{TEXT -1 1 " " }{XPPEDIT 18 0 "s <= 1;" "6#1%\"sG\"\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "As above, what you \+ see here depends on your current definition of " }{TEXT 386 0 "" } {TEXT 387 1 "r" }{TEXT 388 0 "" }{TEXT -1 48 " (which you can change, \+ along with the value of " }{TEXT 389 1 "t" }{TEXT -1 10 " at which " } {TEXT 390 0 "" }{TEXT 391 1 "r" }{TEXT 392 0 "" }{TEXT -1 5 " and " } {TEXT 393 0 "" }{TEXT 394 2 "Dr" }{TEXT 395 0 "" }{TEXT -1 33 " are ev aluated and the range for " }{TEXT 404 1 "t" }{TEXT -1 27 " in the plo t of the curve)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "u:=r(2); v:=Dr(2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p1:=plot([r(t)[1],r(t)[2],t=-3..3]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "p2:=plot([u[1]+v[1]*s,u[2]+v[2]*s,s=0..1], color =blue, thickness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{p 1,p2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Answer questions 4 a nd 5 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Vector-valued functions can also be defined by using the decomp osition into " }{TEXT 396 1 "i" }{TEXT -1 5 " and " }{TEXT 397 1 "j" } {TEXT -1 12 " components." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "i:=[1,0]; j:=[0,1];" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "x:=t->t; y:=t->t^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "r:=t->x(t)*i+y(t)*j;" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 96 "Differentiation can then be carried o ut on the coordinate functions as before. Note the use of " }{TEXT 405 6 "expand" }{TEXT -1 74 " to carry out the vector operations of ad dition and scalar multiplication." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Dr:=t->D(x)(t)*i+D(y)(t)*j; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Dr(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 " For plotting a vector-valued function in this format, it is still conv enient to work directly with the functions " }{XPPEDIT 18 0 "x(t);" "6 #-%\"xG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y(t);" "6#-%\"yG6# %\"tG" }{TEXT -1 42 " as a parametric description of the curve." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Do your c alculations and plots for steps 6 and 7 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 272 26 "4. Arc Length in the \+ Plane" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart; with(linalg): with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We redefine one of ou r curves from Part 3, using the parametric format, and we calculate th e velocity vector as well." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "i:=vector([1,0]); j:=vector([0,1]); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x:=t->cos(t); y:=t->sin(t);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "r:=t->x(t)*i+y(t)*j;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "Dr:=t->diff(x(t),t)*i+diff(y(t),t)*j;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Dr(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "Now we can compute the length of the curve using the arc length formula. Recall that the dot product of a vector with itself i s the square of the length of the vector. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "length_of_Dr:=sqrt (dotprod(Dr(t),Dr(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Noti ce the bars over some of the " }{TEXT 330 1 "t" }{TEXT -1 27 "'s -- an d notice also that " }{TEXT 331 5 "Maple" }{TEXT -1 72 " does not simp lify this expression as we would expect. This is because " }{TEXT 332 6 "Maple'" }{TEXT -1 206 "s \"native\" mode is to work in complex \+ numbers, and those bars represent complex conjugates. Of course, when the numbers are real, the complex conjugate is the same as the number itself, so we need to tell " }{TEXT 333 5 "Maple" }{TEXT -1 6 " that \+ " }{TEXT 334 1 "t" }{TEXT -1 24 " takes only real values." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assum e(t, real):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "length_of_Dr :=sqrt(dotprod(Dr(t),Dr(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Also reca ll the following facts about " }{TEXT 324 5 "Maple" }{TEXT -1 15 "'s i ntegration:" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{TEXT 329 3 "Int" } {TEXT -1 46 " sets up an integral but does not evaluate it." }}{PARA 0 "" 0 "" {TEXT -1 5 "(b) " }{TEXT 328 3 "int" }{TEXT -1 44 " tries t o evaluate an integral symbolically." }}{PARA 0 "" 0 "" {TEXT -1 13 "( c) Whether " }{TEXT 327 3 "int" }{TEXT -1 91 " succeeds or not, the n umerical value of a definite integral can be found by following the " }{TEXT 326 3 "int" }{TEXT -1 14 " command with " }{TEXT 325 9 "evalf(% );" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "arclength:=Int(length_of_Dr,t=0..2*Pi);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "arclength:=int(length_ of_Dr,t=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(% );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Enter your answers for st ep 2 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Do your calculations for the l ength of Pluto's orbit here, and enter your answers for steps 3 throug h 6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "5. " }{TEXT 261 17 "Vectors in Space " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart: with(plots): with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Enter the definitions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "v:=[1,2,-3]; w:=[-7,1,0];\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "We use the same strategy to plot vectors in space as we \+ did to plot vectors in the plane, but the basic plot command is " } {TEXT 262 10 "spacecurve" }{TEXT -1 13 ", and we use " }{TEXT 263 9 "d isplay3d" }{TEXT -1 236 " to show several plots together. After the tw o vectors are displayed, click on the plot, grab it with the mouse, an d rotate it to see the plot from different points of view. Do this wit h all of the later three dimensional plots as well." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "vplot:=sp acecurve([t*u[1],t*u[2],t*u[3],t=0..1], color=green, thickness=2, axes =normal):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "wplot:=spacecurve([t* v[1],t*v[2],t*v[3],t=0..1],thickness=2, color=blue, axes=normal):\ndis play3d(vplot,wplot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Answer step 1 here." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 " \+ Next we'll use " }{TEXT 264 5 "Maple" }{TEXT -1 44 " to calculate dot \+ products. Recall that the " }{TEXT 265 7 "dotprod" }{TEXT -1 24 " comm and is part of the " }{TEXT 266 6 "linalg" }{TEXT -1 44 " package. We' ll display the current vectors " }{TEXT 343 1 "v" }{TEXT -1 5 " and " }{TEXT 267 2 "w " }{TEXT -1 35 "and then calculate the dot product." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "v; w;\ndotprod(v,w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Answer steps \+ 2 through 5 here. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 335 14 "6. Projections" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "Answer questions 2 and 3 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart: with(plots): with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "4." }}{PARA 0 "" 0 "" {TEXT -1 27 "Enter and plot the vectors " }{TEXT 336 1 "v" }{TEXT -1 5 " and " }{TEXT 337 1 "w" }{TEXT -1 138 ". Rotate the figure to get a visual sense of the plane of these two ve ctors, the angle between them, and the length of the projection of " } {TEXT 344 1 "v" }{TEXT -1 6 " onto " }{TEXT 345 1 "w" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "v:=[2,1,3];\nw:=[2,4,2];\nvplot:=spacecurve([t*v[1],t*v[2],t* v[3],t=0..1], color=green, scaling=constrained, thickness=2, axes=norm al):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "wplot:=spacecurve([t*w[1], t*w[2],t*w[3],t=0..1],thickness=2, color=blue, scaling=constrained, ax es=normal):\ndisplay3d(vplot,wplot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The length of " }{TEXT 351 1 "w" }{TEXT -1 46 " is calculated here. Enter the comp onents of " }{TEXT 338 2 "u " }{TEXT -1 34 "in place of the question m arks. [" }{TEXT 352 5 "Maple" }{TEXT -1 20 " note: Don't enter " } {TEXT 353 1 "u" }{TEXT -1 25 " as a scalar multiple of " }{TEXT 354 1 "w" }{TEXT -1 39 " -- you need to give the components of " }{TEXT 355 1 "u" }{TEXT -1 44 " explicitly.] Calculate the dot product of " } {TEXT 339 1 "u" }{TEXT -1 54 " with itself to make sure it has length \+ one. Then plot" }{TEXT 340 2 " u" }{TEXT -1 15 " together with " } {TEXT 341 1 "v" }{TEXT -1 5 " and " }{TEXT 342 1 "w" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sqrt(dotprod(w,w));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u:=[?,?,?];\ndotprod(u,u);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "uplot:=spacecurve([t*u[1],t*u[2],t*u[3],t=0..1],thic kness=6, color=red, scaling=constrained, axes=normal):\ndisplay3d(vplo t,wplot,uplot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Finish step 4 here -- use your for mula to find " }{TEXT 346 1 "p" }{TEXT -1 12 ", calculate " }{TEXT 347 1 "q" }{TEXT -1 6 " from " }{TEXT 348 1 "p" }{TEXT -1 17 ", and ch eck that " }{TEXT 349 1 "p" }{TEXT -1 5 " and " }{TEXT 350 1 "q" } {TEXT -1 19 " are perpendicular." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Answer step 5 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "7. The Cross Product \+ " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Enter the necessary packages." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart: with(plots): with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "1 . Fill in the table for cross products of basic vectors " }{TEXT 356 1 "i" }{TEXT -1 2 ", " }{TEXT 357 1 "j" }{TEXT -1 6 ", and " } {TEXT 358 1 "k" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 " " }{TEXT 359 17 "i j k" }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ " }{TEXT 360 26 "i 0 k -j" }}{PARA 0 "" 0 "" {TEXT -1 17 " " }{TEXT 361 11 "j " }}{PARA 0 "" 0 " " {TEXT -1 17 " " }{TEXT 362 10 "k " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 2 "3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "4. Enter a pair of gene ric spatial vectors and calculate the cross product. (We assume all t he scalar variables are real in order to " }{TEXT 365 26 "avoid comple x conjugates.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "assume(a, real, b, real, c, real, d, real, e, real, f, real):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "v:=[a,b,c];\nw:=[d,e,f];\ncross:=crosspro d(v,w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Calculate the dot product of each vector \+ with " }{TEXT 363 5 "cross" }{TEXT 364 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dotprod(v,cross); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 8 "5. Set " }{XPPEDIT 18 0 "c = 0;" "6#/%\"cG\"\"!" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "f = 0;" "6#/%\"fG\"\"!" }{TEXT -1 79 " in the formulas above to compute the cross product of vectors tha t lie in the " }{TEXT 366 2 "xy" }{TEXT -1 7 "-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "8. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "v:=[2,1,3];\nw:=[2,4,2];\ncross:=crossprod(v,w);\nvl ength:=sqrt(dotprod(v,v));\nwlength:=sqrt(dotprod(w,w));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "costheta:=dotprod(v,w)/(vlength*wlength);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "theta:=arccos(costheta);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Make sure you know what was computed in each line above. Then compare the length of cross with the area of the parallelogram: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sqrt(dotprod(cross,cross));\nvlength*wlength*sin(theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Are they the same? How can you tell?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Repeat the calculation with different vectors \+ " }{TEXT 269 1 "v" }{TEXT -1 5 " and " }{TEXT 270 1 "w" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "8. Arc Length in Space " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Rest art as before." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart: w ith(plots): with(linalg):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Enter the components of your vector-valued functi on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "x:=t->???;\ny:=t->???;\nz:=t->???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "Differentiate all three functions:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Dx:=t->diff(x(t),t);\nDy:=t->diff(y(t),t);\nDz:=t->diff(z(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Then plot the curve using the spacecurve function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "p1:=spacecurve([x(t),y(t),z(t),t=0 ..2*Pi], axes=BOXED, thickness=2, numpoints=800, labels=[\"x axis\",\" y axis\",\"z axis\"], color=black): display(p1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 34 "Here is the arclength computation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Int(sqrt(Dx(t)^2+Dy(t)^2+Dz(t)^2),t =0..2*Pi);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "arclength:=int(sqrt(D x(t)^2+Dy(t)^2+Dz(t)^2),t=0..2*Pi);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Record your ans wers to the questions here." }}{PARA 0 "" 0 "" {TEXT -1 4 "1. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 268 11 "9. Sum mary " }}{PARA 0 "" 0 "" {TEXT -1 49 "Write your answers to the summar y questions here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "7. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "8. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 1 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }