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"" -1 626 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 627 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 628 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 629 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 630 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 631 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 632 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 633 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 634 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 635 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 636 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 637 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 638 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 639 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 640 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 641 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 642 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 643 "" 1 18 128 0 128 1 0 1 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 257 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 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 1 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" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 519 26 " Math 233 Fall 2005 Exam 1" }}{PARA 0 "" 0 "" {TEXT 260 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 5 " Load" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Execute these lines, in the the order they appear:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protect ed names norm and trace have been redefined and unprotected\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }} {PARA 7 "" 1 "" {TEXT -1 46 "Warning, the name distance has been redef ined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "vectorLength := u -> sqrt(sum(u['j']^2, 'j' = 1..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%-vectorLengthGf*6#%\"uG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#-%$sumG 6$*$)&9$6#.%\"jG\"\"#\"\"\"/F7;F:\"\"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "realDotProd := (u,v) -> sum(u['j']*v['j'], 'j' = 1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,realDotProdGf*6$%\"uG%\" vG6\"6$%)operatorG%&arrowGF)-%$sumG6$*&&9$6#.%\"jG\"\"\"&9%F3F6/F4;F6 \"\"$F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "vectorProjec tion := (u,v) -> map(z -> realDotProd(u,v)/realDotProd(v,v)*z, v);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%1vectorProjectionGf*6$%\"uG%\"vG6\"6 $%)operatorG%&arrowGF)-%$mapG6$f*6#%\"zGF)F*F)*(-%,realDotProdG6$T$T& \"\"\"-F56$F8F8!\"\"9$F9F)F)6&F'F=F(9%F?F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 266 1 "1" }{TEXT 357 2 ". " }{TEXT 336 15 "Suppose that " }{TEXT 606 1 "u" }{TEXT 607 7 " \+ and " }{TEXT 608 2 " v" }{TEXT 609 91 " are parallel nonzero vectors . Which of the following statements must be true?\n\n ( I ) " } {TEXT 610 1 "u" }{TEXT 611 1 " " }{TEXT 614 2 " =" }{TEXT 615 1 " " } {XPPEDIT 616 1 "lambda;" "6#%'lambdaG" }{TEXT 612 1 "v" }{TEXT 613 21 " for some scalar " }{XPPEDIT 335 1 "lambda" "6#%'lambdaG" }{TEXT 334 31 " \n( II ) The dot product of " }{TEXT 617 1 "u" }{TEXT 618 6 " and " }{TEXT 619 2 " v" }{TEXT 620 40 " is 0. \n( III ) The cro ss product of " }{TEXT 621 1 "u" }{TEXT 622 6 " and " }{TEXT 623 2 " v" }{TEXT 624 23 " is the zero vector. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 143 "a) Not one of the three sta tements must be true\nb) I only \nc) II only \nd) III \+ only \ne) I and II only\nf) I and III only" }}{PARA 258 "" 0 " " {TEXT 337 110 "g) II and III only \nh) All three statements mus t be true \ni) Wong answer\nj) Bonus wrong answer" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 605 18 "Solution: f " }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT 292 4 "2. " }{TEXT -1 39 "Suppose that the magnitude of ve ctor " }{XPPEDIT 19 1 "v = `<`*1,-8,c*`>`;" "6%/%\"vG*&%\"GF'" }{TEXT -1 23 " is 9 and that \+ " }{XPPEDIT 19 1 "v " "6#%\"vG" }{TEXT -1 28 " is parallel to \n \+ to " }{XPPEDIT 19 1 "`<`*a,b,-12*`>`;" "6%*&%\"GF%!\"\"" }{TEXT -1 15 " . What is " }{XPPEDIT 19 1 " c;" "6#%\"cG" }{TEXT -1 3 " ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT 338 4 "a) " }{XPPEDIT 339 1 "0" "6#\"\"!" } {TEXT 340 9 " b) " }{XPPEDIT 341 1 "1" "6#\"\"\"" }{TEXT 342 10 " c) " }{XPPEDIT 343 1 "-1" "6#,$\"\"\"!\"\"" }{TEXT 344 7 " d ) " }{XPPEDIT 345 1 "2" "6#\"\"#" }{TEXT 346 10 " e) " } {XPPEDIT 347 1 "-2" "6#,$\"\"#!\"\"" }{TEXT 348 11 " f) " } {XPPEDIT 349 1 "3" "6#\"\"$" }{TEXT 350 11 " g) " }{XPPEDIT 351 1 "-3" "6#,$\"\"$!\"\"" }{TEXT 352 8 " h) " }{XPPEDIT 353 1 "4 " "6#\"\"%" }{TEXT 354 8 " i) " }{XPPEDIT 355 1 "-4" "6#,$\"\"%!\" \"" }{TEXT 356 76 " \nj) Insufficient informa tion to determine answer " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 574 17 "Solution: j " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can see that " }{XPPEDIT 19 1 "c = 4;" "6#/%\"cG \"\"%" }{TEXT -1 9 " or " }{XPPEDIT 19 1 "c = -4;" "6#/%\"cG,$\" \"%!\"\"" }{TEXT -1 56 " but we cannot determine which one of these \+ c must be." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v := [1,-8,c];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%\"\"\"!\")%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn := vectorLength(v) = 9;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/*$,&\"#l\"\"\"*$)%\"cG\"\"#F)F )#F)F-\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(eqn, c );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"%\"\"%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "v1 := [1,-8, 4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G7%\"\"\"!\")\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v2 := [1,-8,-4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G7%\"\" \"!\")!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "[a,b,-12] = - 3*v1*` is true with a = -3 and b = 24`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/7%%\"aG%\"bG!#7,$*(\"\"$\"\"\"7%F+!\")\"\"%F+%@~is~true~with~a~ =~-3~and~b~=~24GF+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " [a,b,-12] = 3*v2*` is true with a = 3 and b = -24`;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/7%%\"aG%\"bG!#7,$*(\"\"$\"\"\"7%F+!\")!\"%F+%@~is~tr ue~with~a~=~3~and~b~=~-24GF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 263 18 "3. If the vectors " }{TEXT 361 2 " " }{XPPEDIT 364 1 "`<`*(a+b),a+3, (-b)*`>`;" "6%*&%\"GF%" }{TEXT 358 1 " " }{TEXT 362 3 "and" }{TEXT 363 2 " " } {XPPEDIT 365 1 "`<`*(a-b),-4,(-b)*`>`;" "6%*&%\"GF%" }{TEXT 359 1 " " }{TEXT 366 41 " ar e perpendicular, \n then what might " }{TEXT 367 1 " " }{XPPEDIT 368 1 "a;" "6#%\"aG" }{TEXT 360 2 " " }{TEXT 369 4 "be? " }{TEXT 370 2 " " }}{PARA 3 "" 0 "" {TEXT 291 5 "a) " }{TEXT 527 1 "-" }{TEXT 528 16 " 5 b) " }{TEXT 525 1 "-" }{TEXT 526 19 " 4 \+ c) " }{TEXT 529 1 "-" }{TEXT 530 19 " 3 d) " }{TEXT 531 1 "-" }{TEXT 532 17 " 2 e) " }{TEXT 533 1 "-" }{TEXT 534 99 " 1 \nf) 1 g) 2 h) 3 \+ i) 4 j) 5 " }{TEXT 371 7 " \n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 576 13 "Solution : d" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "expression := realDotProd([a+b,a+3,-b],[a-b,-4,-b]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+expressionG,**&,&%\"aG\"\"\"%\"bGF)F),&F(F)F*!\"\"F) F)*&\"\"%F)F(F)F,\"#7F,*$)F*\"\"#F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn := expand( expression ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*$)%\"aG\"\"#\"\"\"F+*&\"\"%F+F)F+!\"\"\"#7F. \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve( eqn, a );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"'!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Either of these might be the value of " }{XPPEDIT 19 1 "a;" "6#%\"a G" }{TEXT -1 43 ", but only one appears in the answer list." }}{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 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 1 " " }{TEXT 378 50 "4. What is the cosine of the angle between vectors" }{TEXT 379 3 " " } {XPPEDIT 376 1 "`<`*(-7),-4,4*`>`;" "6%*&%\"GF%" }{TEXT 372 1 " " }{TEXT 374 5 " and" }{TEXT 375 4 " " }{XPPEDIT 377 1 "`<`*1,4,8*`>`;" "6%*&%\"GF%" }{TEXT 373 3 " ?" }}{PARA 0 "" 0 "" {TEXT 270 4 "a) \+ " }{XPPEDIT 256 0 "1/12;" "6#*&\"\"\"F$\"#7!\"\"" }{TEXT 257 13 " \+ " }{TEXT 380 3 "b) " }{TEXT 381 1 " " }{XPPEDIT 297 0 "1/9;" " 6#*&\"\"\"F$\"\"*!\"\"" }{TEXT 298 17 " c) " }{XPPEDIT 299 0 "1/8;" "6#*&\"\"\"F$\"\")!\"\"" }{TEXT 300 7 " " }{TEXT 382 10 " d) " }{XPPEDIT 301 0 "1/6;" "6#*&\"\"\"F$\"\"'!\"\"" } {TEXT 302 5 " " }{TEXT 383 12 " e)" }{TEXT 384 3 " " } {XPPEDIT 305 0 "1/5;" "6#*&\"\"\"F$\"\"&!\"\"" }{TEXT 306 4 " " }} {PARA 257 "" 0 "" {TEXT 261 4 "f) " }{TEXT 258 2 " " }{XPPEDIT 303 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 304 22 " g) \+ " }{XPPEDIT 307 0 "2/9;" "6#*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT 308 22 " \+ h) " }{XPPEDIT 309 0 "5/12;" "6#*&\"\"&\"\"\"\"#7!\" \"" }{TEXT 310 19 " i) " }{XPPEDIT 311 0 "3/8;" "6#*&\" \"$\"\"\"\"\")!\"\"" }{TEXT 259 23 " j) " } {XPPEDIT 313 0 "2/5;" "6#*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT 314 1 " " } {TEXT 312 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 575 19 "Solution: b " } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A := [-7,-4,4]; B := [1,4,8];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7%!\"(!\"%\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG7%\"\"\"\"\"%\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "realDotProd(A,B)/vectorLength(A)/vectorLength(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 271 2 " 5" }{TEXT -1 2 ". " }{TEXT 385 35 "What is the radius of the sphere " }{XPPEDIT 522 1 "x^2+y^2+z^2-10*x+6*y-14*z-38 = 0;" "6#/,0*$%\"xG\"\"#\"\"\"*$% \"yGF'F(*$%\"zGF'F(*&\"#5F(F&F(!\"\"*&\"\"'F(F*F(F(*&\"#9F(F,F(F/\"#QF /\"\"!" }{TEXT 521 2 " " }{TEXT 387 1 " " }{TEXT 386 1 "?" }}{PARA 261 "" 0 "" {TEXT -1 59 "a) 6 b) 7 c) 8 d) 9 e) 10" }}{PARA 261 "" 0 "" {TEXT -1 57 "f) 11 g) 12 \+ h) 13 i) 14 j) 15" }}{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 535 13 "Solution: f" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eqn := x^2+y^2+z^2-10*x +6*y-14*z-38 = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,0*$)%\"x G\"\"#\"\"\"F+*$)%\"yGF*F+F+*$)%\"zGF*F+F+*&\"#5F+F)F+!\"\"*&\"\"'F+F. F+F+*&\"#9F+F1F+F4\"#QF4\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn := completesquare(eqn, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$eqnG/,.*$),&%\"xG\"\"\"\"\"&!\"\"\"\"#F+F+\"#jF-*$)%\"yGF.F+F+*$) %\"zGF.F+F+*&\"\"'F+F2F+F+*&\"#9F+F5F+F-\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn := completesquare(eqn, y); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,,*$),&%\"yG\"\"\"\"\"$F+\"\"#F+F+\"#s!\" \"*$),&%\"xGF+\"\"&F/F-F+F+*$)%\"zGF-F+F+*&\"#9F+F7F+F/\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn := completesquare(eqn, z );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,**$),&%\"zG\"\"\"\"\"(! \"\"\"\"#F+F+\"$@\"F-*$),&%\"yGF+\"\"$F+F.F+F+*$),&%\"xGF+\"\"&F-F.F+F +\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn := map(z -> z + 121, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*$),&%\"zG\" \"\"\"\"(!\"\"\"\"#F+F+*$),&%\"yGF+\"\"$F+F.F+F+*$),&%\"xGF+\"\"&F-F.F +F+\"$@\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 267 7 "6. Let " }{TEXT 396 1 " \+ " }{TEXT 537 3 "p =" }{TEXT 538 2 " " }{XPPEDIT 262 1 "`<`*a,b,c*`>`; " "6%*&%\"GF%" }{TEXT 388 1 " " } {TEXT 397 29 " be the vector projection of " }{TEXT 398 2 " " }{TEXT 539 3 "u =" }{TEXT 540 1 " " }{XPPEDIT 389 1 "`<`*3,-6,12*`>`;" "6%*&% \"GF%" }{TEXT 390 2 " " } {TEXT 399 3 " on" }{TEXT 400 3 " " }{TEXT 541 3 "v =" }{TEXT 542 1 " " }{XPPEDIT 391 1 "`<`*4,-1,2*`>`;" "6%*&%\"GF%" }{TEXT 392 1 " " }{TEXT 401 15 ". \n What is" } {TEXT 402 1 " " }{TEXT 394 1 " " }{XPPEDIT 395 0 "a" "6#%\"aG" }{TEXT 393 2 " ?" }}{PARA 0 "" 0 "" {TEXT 268 132 "\na) 1 b) 2 \+ c) 3 d) 4 e) 5 \n\n f) 6 g) 7 \+ h) 8 i) 9 j) 10" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 536 13 "Solution : h" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "u := [3,-6,12]; v := [4,-1,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"$!\"'\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%\"\"%!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p := vectorProjection( u , v );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG7%\"\")!\"#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q := u-p; #line 1 of verification" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"qG7%!\"&!\"%\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dotprod( p , q ); #lin e 2 of verification" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {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 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 269 8 "7. Let " }{TEXT 548 1 "u" }{TEXT 545 4 " , " }{TEXT 549 1 "v" }{TEXT 550 7 ", and " }{TEXT 546 1 "p" }{TEXT 547 41 " be as in the preceding question. Let " }{TEXT 543 1 "q" } {TEXT 544 2 " =" }{TEXT 406 1 " " }{XPPEDIT 410 1 "`<`*d,e,f*`>`;" "6% *&%\"GF%" }{TEXT 411 1 " " }{TEXT 407 48 " be the vector \n that is perpendicular to " }{TEXT 553 1 "p" }{TEXT 554 17 " and such that " }{TEXT 552 9 "u = p + q" } {TEXT 551 4 " . " }{TEXT 408 1 " " }{TEXT 409 7 "What is" }{TEXT 404 2 " " }{TEXT 555 1 "d" }{TEXT 556 1 " " }{TEXT 403 1 "?" }{TEXT 405 1 "\n" }}{PARA 0 "" 0 "" {TEXT 412 128 "a) -5 b) -4 c ) -3 d) -2 e) -1 \n\n f) 0 g) 1 h) 2 i) 3 j) 4" }}{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 557 13 "Solution: a " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "(See preceding question for calculation.)" }}{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 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 430 9 "8. Let " } {XPPEDIT 422 1 "`<`*a,b,c*`>`" "6%*&%\"GF%" }{TEXT 423 1 " " }{TEXT 418 28 " be the vector cross product" }{TEXT 428 1 " " }{XPPEDIT 426 1 "`<`*2,1,1*`>` " "6%*&%\"GF%" }{TEXT 427 1 " " }{TEXT 429 2 " " }{TEXT 424 1 "X " }{TEXT 425 1 " " }{TEXT 419 1 " " }{XPPEDIT 413 1 "`<`*4,-3,1*`>`;" "6%*&%\"GF%" }{TEXT 414 1 " " } {TEXT 420 9 ". What is" }{TEXT 421 1 " " }{TEXT 416 1 " " }{XPPEDIT 417 0 "a+b+c" "6#,(%\"aG\"\"\"%\"bGF%%\"cGF%" }{TEXT 415 2 " ?" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 431 128 "a) -5 b) -4 c) -3 d) -2 \+ e) -1 \n\n f) 0 g) 1 h) 2 i) 3 \+ j) 4" }}{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 643 16 "Solution: b " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "cp := crossp rod( [2,1,1] , [4, -3, 1] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG -%'vectorG6#7%\"\"%\"\"#!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cp[1]+cp[2]+cp[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}} {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 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 54 "9. What is the \+ sine of the angle between the vectors " }{XPPEDIT 432 1 "`<`*2,-4,-4* `>`;" "6%*&%\"GF%F)" }{TEXT 433 7 " and " }{XPPEDIT 434 1 "`<`*1,2,2*`>`;" "6%*&%\"GF%" }{TEXT 435 2 " ?" }{TEXT 436 1 " " }}{PARA 258 "" 0 "" {TEXT 437 5 "\na) " }{XPPEDIT 438 1 "sqrt(2)/3;" "6#*&-%%sqrtG6# \"\"#\"\"\"\"\"$!\"\"" }{TEXT 439 9 " b) " }{XPPEDIT 440 1 "sqrt( 2)/6;" "6#*&-%%sqrtG6#\"\"#\"\"\"\"\"'!\"\"" }{TEXT 441 10 " c) \+ " }{XPPEDIT 442 1 "sqrt(2)/9;" "6#*&-%%sqrtG6#\"\"#\"\"\"\"\"*!\"\"" } {TEXT 443 8 " d) " }{XPPEDIT 524 1 "2*sqrt(2)/3;" "6#*(\"\"#\"\"\" -%%sqrtG6#F$F%\"\"$!\"\"" }{TEXT 523 8 " e) " }{XPPEDIT 444 1 "2*s qrt(2)/9;" "6#*(\"\"#\"\"\"-%%sqrtG6#F$F%\"\"*!\"\"" }{TEXT 445 9 " \+ f) " }{XPPEDIT 446 1 "4*sqrt(2)/3;" "6#*(\"\"%\"\"\"-%%sqrtG6#\"\"# F%\"\"$!\"\"" }{TEXT 447 11 " g) " }{XPPEDIT 448 1 "4*sqrt(2)/9 ;" "6#*(\"\"%\"\"\"-%%sqrtG6#\"\"#F%\"\"*!\"\"" }{TEXT 449 8 " h) \+ " }{XPPEDIT 450 1 "sqrt(2)/2;" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\"" } {TEXT 451 8 " i) " }{XPPEDIT 452 1 "sqrt(2)/4;" "6#*&-%%sqrtG6#\" \"#\"\"\"\"\"%!\"\"" }{TEXT 453 76 " \nj) Ins ufficient information to determine answer " }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 577 14 "Solution: g " }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "u := [2,-4,-4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"uG7%\"\"#!\"%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "v : = [1,2,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%\"\"\"\"\"#F'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A := crossprod([2,-4,-4],[ 1,2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'vectorG6#7%\"\"!! \")\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "vectorLength(A) /vectorLength(u)/vectorLength(v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*(\"\"%\"\"\"\"\"*!\"\"\"\"##F&F)F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 602 1 " " }{TEXT 273 38 "10. What is the area of parallelogram " }{TEXT 454 5 " PQRS" }{TEXT 455 7 " if " }{TEXT 456 1 "P" }{TEXT 457 16 " = ( 4, 3, 5 ), " }{TEXT 458 2 " Q" }{TEXT 459 22 " = ( 1, 2, 3 ), and " }{TEXT 460 1 "R" }{TEXT 461 16 " = ( 3 , 4, 4 )?\n" }{TEXT 603 0 "" }{TEXT 315 28 "\na) 4 b) " }{XPPEDIT 316 0 "3*sqrt(2)" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" } {TEXT 317 16 " c) " }{XPPEDIT 318 0 "2*sqrt(6)" "6#*&\"\"# \"\"\"-%%sqrtG6#\"\"'F%" }{TEXT 319 20 " d) " } {XPPEDIT 320 0 "5" "6#\"\"&" }{TEXT 321 23 " e) " } {XPPEDIT 322 0 "sqrt(26);" "6#-%%sqrtG6#\"#E" }{TEXT 323 21 " \+ \nf) " }{XPPEDIT 324 0 "4*sqrt(2);" "6#*&\"\"%\"\"\"-%%sqrtG6# \"\"#F%" }{TEXT 325 14 " g) " }{XPPEDIT 326 0 "sqrt(42)" "6# -%%sqrtG6#\"#U" }{TEXT 327 18 " h) " }{XPPEDIT 328 0 "4* sqrt(3)" "6#*&\"\"%\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 329 21 " \+ i) " }{XPPEDIT 330 0 "2*sqrt(13)" "6#*&\"\"#\"\"\"-%%sqrtG6#\"# 8F%" }{TEXT 331 17 " j) " }{XPPEDIT 332 0 "sqrt(58)" "6#- %%sqrtG6#\"#e" }}{PARA 0 "" 0 "" {TEXT 333 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 604 15 "Solution: e" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "P := [4,3,5]; Q := [1,2, 3]; R := [3,4,4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG7%\"\"%\" \"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG7%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7%\"\"$\"\"%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PQ := Q-P; PR := R-P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PQG7%!\"$!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PRG7%!\"\"\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "CP := crossprod(PQ,PR);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CPG-%'vectorG6#7%\"\"$!\"\"!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vectorLength(CP);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *$\"#E#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 274 76 "11. What is the volume of the \+ parallelepiped formed from the three vectors" }{TEXT 468 1 " " } {XPPEDIT 462 1 "`<`*2,-1,1*`>`;" "6%*&%\"GF%" }{TEXT 463 3 ", " }{XPPEDIT 464 1 "`<`*2,1,1*`>` " "6%*&%\" GF%" }{TEXT 465 3 ", " }{TEXT 470 3 "and" }{TEXT 471 2 " " }{XPPEDIT 466 1 "`<`*1,2,3*`>` " "6%*&%\"GF%" }{TEXT 467 1 "?" }{TEXT 469 3 " \n\n" }{TEXT 296 79 "a) 1 b) 2 c) 4 d) 6 \+ e) 8" }{TEXT 560 21 " " }}{PARA 0 "" 0 "" {TEXT 473 36 "f) 10 g) 12 " }{TEXT -1 1 " " }{TEXT 558 2 "h)" }{TEXT 472 37 " 16 i) 20 j ) " }{TEXT 559 2 "24" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 520 14 "Solution: f " }{TEXT -1 1 "\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "u := vector([2,-1,1]); v := vector([2,1,1]); w := vector([1,2,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'vectorG6#7%\"\"#!\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7%\"\"#\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'vectorG6#7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "abs( dotprod(u,crossprod(v,w)) );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{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 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 275 340 "12. Consider the following two \+ lines: (i) the line through the points ( 3, 0, -6 ) and ( 7, \+ 8, 6 ), and \n (ii) the line with symmetric equations (x - 1 )/2 = y - 2 = z + 2. The two lines \n\na) are parallel \nb ) do not intersect but are not parallel \nc) intersect \+ at a point whose x-coordinate is " }{TEXT 629 1 "-" }{TEXT 630 60 " 2 \nd) intersect at a point whose x-coordinate is " }{TEXT 631 3 " - " }{TEXT 632 63 "1 \ne) intersect at a point who se x-coordinate is 0" }{TEXT 475 21 " " }}{PARA 0 "" 0 "" {TEXT 476 5 "f) " }{TEXT 633 1 " " }{TEXT 635 42 "intersec t at a point whose x-coordinate is" }{TEXT 634 20 " 1 \ng) \+ " }{TEXT 636 42 "intersect at a point whose x-coordinate is" }{TEXT 637 16 " 2 " }{TEXT -1 2 " \n" }{TEXT 477 5 "h) " } {TEXT 638 42 "intersect at a point whose x-coordinate is" }{TEXT 639 21 " 3 \ni) " }{TEXT 640 42 "intersect at a point whose \+ x-coordinate is" }{TEXT 641 22 " 4 \nj) " }{TEXT -1 1 " " }{TEXT 642 42 "intersect at a point whose x-coordinate is" }{TEXT 478 4 " 5" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT 474 1 " " }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 601 16 "Solution: j " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P := [ 3, 0, -6]: \nQ := [7, 8, 6]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PQ := Q-P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PQG7 %\"\"%\"\")\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "PQ := (1 /4)*PQ; #get smaller numbers to work with" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PQG7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "param1 := x = P[1]+t*PQ[1];\nparam2 := y = P[2]+t*PQ [2];\nparam3 := z = P[3]+t*PQ[3]; #parametric equations of first line " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'param1G/%\"xG,&\"\"$\"\"\"%\"tG F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'param2G/%\"yG,$*&\"\"#\"\"\"% \"tGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'param3G/%\"zG,&\"\"'!\" \"*&\"\"$\"\"\"%\"tGF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "eqn1 := (x - 1)/2 = y - 2; # first of the given symmetric equations" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&*&\"\"#!\"\"%\"xG\"\"\"F+ #F+F(F),&%\"yGF+F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "eqn 2 := y - 2 = z + 2; # second of the given symmetric equations" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&%\"yG\"\"\"\"\"#!\"\",&%\"z GF(F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn3 := subs( \{ param1,param2\} , eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/, &\"\"\"F'*&\"\"#!\"\"%\"tGF'F',&*&F)F'F+F'F'F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn4 := subs( \{param2,param3\} , eqn2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/,&*&\"\"#\"\"\"%\"tGF)F)F(!\" \",&\"\"%F+*&\"\"$F)F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "solve(eqn3, t); solve(eqn4,t); #intersection iff same answer" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(t=2,param1 ); # The answer" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "subs(t=2, [P[1]+t*PQ[1],P[2] +t*PQ[2],P[3]+t*PQ[3]]); #intersection point" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"&\"\"%\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 276 1 " " }{TEXT 483 17 "13. The vector " }{TEXT 479 1 "<" }{TEXT 480 11 " 4 , b , c " }{TEXT 481 1 ">" }{TEXT 482 71 " is perpendicular to the plane z = 2x - y + 1. What is b ? " } {TEXT 484 3 " \n\n" }{TEXT 485 132 "a) -4 b) - 3 c) - 2 d) - 1 e) 0 \nf) 1 g) 2 h) 3 \+ i) 4 j) 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 600 18 "Solution: \+ c " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn := [4, b,c] = lambda*[2,-1,-1] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/ 7%\"\"%%\"bG%\"cG*&%'lambdaG\"\"\"7%\"\"#!\"\"F/F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eqn2 := lhs(eqn)[1] = expand(rhs(eqn))[1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"%,$*&\"\"#\"\"\"%'lam bdaGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(eqn2, lam bda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "subs(lambda=2, expand(rhs(eqn)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"%!\"#F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 142 "14. A pl ane contains the point (1, 1, 0) and the line that is the graph of th e\n symmetric equations x/2 = y - 2 = z/2. The Cartesian " } {TEXT 486 67 "equation of the plane is \n Ax + 2y + Cz = D. W hat is D? " }{TEXT 489 1 "\n" }}{PARA 3 "" 0 "" {TEXT 488 132 "a) -4 b) - 3 c) - 2 d) - 1 e) 0 \nf) 1 \+ g) 2 h) 3 i) 4 j) 5" }{TEXT 487 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 599 18 "Solution: i " }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "P := [1, 1, 0]; # Given poin t on plane " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG7%\"\"\"F&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Q := [0, 2, 0]; # Point o n given line, hence on plane " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"Q G7%\"\"!\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "u := [2, 1,2]; # Vector parallel to line, hence parallel to plane" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"#\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "PQ := Q-P; # vector that lies in plane" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PQG7%!\"\"\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "N := crossprod( PQ , u );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'vectorG6#7%\"\"#F)!\"$" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 64 "eqn := N[1]*(x-1) + N[2]*(y-1) + N[3]*(z-0) \+ = 0; # eqtn of plane" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,**&\" \"#\"\"\"%\"xGF)F)\"\"%!\"\"*&F(F)%\"yGF)F)*&\"\"$F)%\"zGF)F,\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn := map( u -> u + 4, eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*&\"\"#\"\"\"%\"xGF)F) *&F(F)%\"yGF)F)*&\"\"$F)%\"zGF)!\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "subs(\{x=P[1], y = P[2], z = P[3]\}, eqn); #first l ine of verification" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"%F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "subs(\{x=2*s, y = 2+s, z = 2 *s\}, eqn); #second line of verification" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"%F$" }}}{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 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 492 107 "15. A plane is perpendicular to the line 1 - x = y/3 = 2z . The plane passes thr ough the \n point ( " }{TEXT 627 1 "-" }{TEXT 628 4 " 5, " }{TEXT 625 1 "-" }{TEXT 626 56 "1, 0 ) as well as the point ( 0, 0, c ). W hat is c ?" }{TEXT 493 3 " \n" }{TEXT 491 133 "\na) -4 b) - 3 c) - 2 d) - 1 e) 0 \nf) 1 g) 2 h) 3 i) 4 j) 5" }{TEXT 490 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 598 17 "Solution: \+ i " }{TEXT -1 0 "" }}{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 50 "eqn := -(x - (-5)) + 3*(y - (-1)) + (z - 0)/2 = 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,*%\"xG!\"\"\"\"#F(*&\"\"$ \"\"\"%\"yGF,F,*&F)F(%\"zGF,F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "solve( subs( \{x=0, y=0, z=c\}, eqn ) , c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{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 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 265 120 "16. The \+ two planes x - y + z = 6 and 2x + y - 4z = 0 intersect \+ in a line that is parallel to the \nvector" }{TEXT 501 2 " " }{TEXT 496 1 "<" }{TEXT 497 11 " 1 , b , c " }{TEXT 498 1 ">" }{TEXT 499 16 " . What is b ? " }{TEXT 500 3 " \n\n" }{TEXT 495 132 "a) -4 \+ b) - 3 c) - 2 d) - 1 e) 0 \nf) 1 g) 2 h) 3 i) 4 j) 5" }{TEXT 494 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 597 17 "Solution: g " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "cp := crossprod([1,-1,1],[2,1,-4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'vectorG6#7%\"\"$\"\"'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(1/3*cp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"\"\"\"#F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 13 "17. A plane " }{XPPEDIT 19 1 "P;" "6#% \"PG" }{TEXT -1 73 " is parallel to the plane x - y + 2z = 0. The po int (3, 1, 2) lies on " }{XPPEDIT 19 1 "P;" "6#%\"PG" }{TEXT -1 50 " \+ . What is the sum of the three intercepts that " }{XPPEDIT 19 1 "P; " "6#%\"PG" }{TEXT -1 31 " has with the coordinate axes?" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 156 "a) 3 \+ b) 4 c) 5 d) 6 e) 7 \+ \nf) 8 g) 9 h) 10 i) 10 j) 12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 572 6 "Answer" }{TEXT 573 6 ": a " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "eqn := (x-3)-(y-1)+2*(z-2)=0 ; # Cartesian equation of P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqn G/,*%\"xG\"\"\"\"\"'!\"\"%\"yGF**&\"\"#F(%\"zGF(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "solve( subs(\{y=0,z=0\}, eqn) , x) \n + solve( subs(\{x=0,z=0\}, eqn) , y)\n + solve( subs(\{x=0,y= 0\}, eqn) , z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 233 "18. Examine the equation s in each of answers (a) through (h). Determine which, if any, of the se equations has a graph with circles for level curves and whose slice s with planes parallel to the xz-plane are hyperbolas. If there is " }{TEXT 293 11 "exactly one" }{TEXT 294 1 " " }{TEXT -1 116 "such equat ion, choose that as your answer. Otherwise, choose (i) or (j), whichev er is appropriate, as your answer. \n" }}{PARA 261 "" 0 "" {TEXT -1 4 "a) " }{TEXT 504 1 " " }{XPPEDIT 505 1 "x^2+4*x+y^2+z^2-2*z=0" "6#/,, *$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(F(*$%\"yGF'F(*$%\"zGF'F(*&F'F(F.F(!\"\" \"\"!" }{TEXT 503 2 " " }{TEXT -1 22 " b) " }{TEXT 507 1 " " }{XPPEDIT 508 1 "x^2+ y^2 - z^2 =0" "6#/,(*$%\"xG\"\"#\"\" \"*$%\"yGF'F(*$%\"zGF'!\"\"\"\"!" }{TEXT -1 6 " \nc) " }{TEXT 509 1 " " }{XPPEDIT 510 1 "x^2+ y^2 - z^2 = 1" "6#/,(*$%\"xG\"\"#\"\"\"*$%\" yGF'F(*$%\"zGF'!\"\"F(" }{TEXT -1 44 " \+ d) " }{XPPEDIT 511 1 "x^2 - y^2 - z^2 = 1" "6#/,(*$%\"xG\"\"# \"\"\"*$%\"yGF'!\"\"*$%\"zGF'F+F(" }{TEXT -1 8 " \ne) " }{XPPEDIT 513 1 "x^2+ y^2 - z= 1" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(%\"zG!\"\" F(" }{TEXT -1 47 " f) " } {XPPEDIT 515 1 "x^2+ y^2 - z = 0" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F (%\"zG!\"\"\"\"!" }{TEXT 516 1 "\n" }{TEXT -1 6 "g) " }{XPPEDIT 517 1 "x^2-y^2-z^2 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"*$%\"zG F'F+\"\"!" }{TEXT -1 43 " h) " }{XPPEDIT 518 1 "x^2 - y^2 + z= 0" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'! \"\"%\"zGF(\"\"!" }{TEXT -1 3 " " }{TEXT 295 1 "\n" }{TEXT -1 62 "i) not one of the equations above has the required properties" }}{PARA 261 "" 0 "" {TEXT -1 69 "j) more than one of the equations above have the required properties" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 561 14 "Solution: c " }{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 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 571 39 "19. A point has spherical coordinates \+ " }{XPPEDIT 580 0 "rho = 6;" "6#/%$rhoG\"\"'" }{TEXT 579 8 " and " }{XPPEDIT 581 0 "phi = Pi/3;" "6#/%$phiG*&%#PiG\"\"\"\"\"$!\"\"" } {TEXT 578 58 ". \n What is the sum of it cylindrical coordinate s " }{XPPEDIT 582 0 "r;" "6#%\"rG" }{TEXT 583 9 " and " } {XPPEDIT 585 0 "z;" "6#%\"zG" }{TEXT 584 3 " ?" }{TEXT -1 4 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 586 7 "a) \+ " }{XPPEDIT 18 0 "2+sqrt(2);" "6#,&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 587 18 " b) " }{XPPEDIT 18 0 "2+2*sqrt(2);" "6#,&\"\"#\" \"\"*&F$F%-%%sqrtG6#F$F%F%" }{TEXT 588 23 " c) " } {XPPEDIT 18 0 "2+3*sqrt(2);" "6#,&\"\"#\"\"\"*&\"\"$F%-%%sqrtG6#F$F%F% " }{TEXT 589 17 " \nd) " }{XPPEDIT 18 0 "3+sqrt(3);" "6#,& \"\"$\"\"\"-%%sqrtG6#F$F%" }{TEXT 590 18 " e) " } {XPPEDIT 18 0 "3+2*sqrt(3);" "6#,&\"\"$\"\"\"*&\"\"#F%-%%sqrtG6#F$F%F% " }{TEXT 591 24 " f) " }{XPPEDIT 18 0 "3+3*sqrt(3) ;" "6#,&\"\"$\"\"\"*&F$F%-%%sqrtG6#F$F%F%" }{TEXT 592 13 " \ng) \+ " }{XPPEDIT 18 0 "sqrt(2)+sqrt(3);" "6#,&-%%sqrtG6#\"\"#\"\"\"-F%6# \"\"$F(" }{TEXT 593 15 " h) " }{XPPEDIT 18 0 "2*sqrt(2)+3*s qrt(3);" "6#,&*&\"\"#\"\"\"-%%sqrtG6#F%F&F&*&\"\"$F&-F(6#F+F&F&" } {TEXT 594 18 " i) " }{XPPEDIT 18 0 "3*sqrt(2)+2*sqrt(3); " "6#,&*&\"\"$\"\"\"-%%sqrtG6#\"\"#F&F&*&F*F&-F(6#F%F&F&" }{TEXT 595 66 " \n j) Insufficient information to determine the answe r" }}{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 596 17 "Solution: f " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rho := 6: phi := Pi/3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "z := rho*cos(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"zG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r := rho*si n(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG,$*&\"\"$\"\"\"F'#F( \"\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "z+r;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"$\"\"\"*&F$F%F$#F%\"\"#F%" }}}{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 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 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 277 34 "20. The set that is described by " }{XPPEDIT 506 1 "2*z = r^2 ,` `*0 < r;" "6$/*&\"\"#\"\"\"%\"zGF&*$%\"rGF%2*&%#~~GF&\"\"!F&F)" } {TEXT 502 116 " in cylindrical coordinates is described by which of t he following equations in spherical coordinates for " } {XPPEDIT 567 1 "0 < phi;" "6#2\"\"!%$phiG" }{TEXT 564 2 " " } {XPPEDIT 566 1 "`` < Pi/2;" "6#2%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 565 9 " ? \n" }}{PARA 3 "" 0 "" {TEXT 278 4 "a) " }{XPPEDIT 279 0 "rho = 2*csc(phi);" "6#/%$rhoG*&\"\"#\"\"\"-%$cscG6#%$phiGF'" } {TEXT 280 30 " b) " }{XPPEDIT 281 0 "rho=2*s in(phi)" "6#/%$rhoG*&\"\"#\"\"\"-%$sinG6#%$phiGF'" }{TEXT 282 18 " \+ \nc) " }{XPPEDIT 283 0 "rho = 2*sec(phi);" "6#/%$rhoG*&\"\"# \"\"\"-%$secG6#%$phiGF'" }{TEXT 284 30 " d) \+ " }{XPPEDIT 285 0 "rho=2*cos(phi)" "6#/%$rhoG*&\"\"#\"\"\"-%$cosG6#%$p hiGF'" }{TEXT 286 17 " \ne) " }{XPPEDIT 287 0 "rho = 2*cot (phi);" "6#/%$rhoG*&\"\"#\"\"\"-%$cotG6#%$phiGF'" }{TEXT 288 31 " \+ f) " }{XPPEDIT 289 0 "rho = 2*tan(phi);" "6#/%$ rhoG*&\"\"#\"\"\"-%$tanG6#%$phiGF'" }{TEXT 290 17 " \ng) \+ " }{XPPEDIT 257 0 "rho = 2*cot(phi)*csc(phi);" "6#/%$rhoG*(\"\"#\"\"\" -%$cotG6#%$phiGF'-%$cscG6#F+F'" }{TEXT 562 17 " h) " } {XPPEDIT 256 0 "rho = 2*tan(phi)*sec(phi);" "6#/%$rhoG*(\"\"#\"\"\"-%$ tanG6#%$phiGF'-%$secG6#F+F'" }{TEXT 563 4 " " }{TEXT 512 6 " \+ " }{TEXT 514 9 " \n i) " }{XPPEDIT 256 0 "rho = 2*cot(phi)*cos(phi) ;" "6#/%$rhoG*(\"\"#\"\"\"-%$cotG6#%$phiGF'-%$cosG6#F+F'" }{TEXT 568 18 " j) " }{XPPEDIT 256 0 "rho = 2*tan(phi)*sin(phi);" " 6#/%$rhoG*(\"\"#\"\"\"-%$tanG6#%$phiGF'-%$sinG6#F+F'" }{TEXT 569 5 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 570 14 "Solutio n: g " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eqn := \+ 2*z = r^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*&\"\"#\"\"\"% \"zGF)F)*$)%\"rGF(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn := subs( r = sqrt(x^2+y^2), eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$eqnG/,$*&\"\"#\"\"\"%\"zGF)F),&*$)%\"xGF(F)F)*$)%\"yGF(F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn := map( u -> u + z^2, eq n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&*&\"\"#\"\"\"%\"zGF)F )*$)F*F(F)F),(*$)%\"xGF(F)F)*$)%\"yGF(F)F)F+F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "eqn := completesquare(eqn,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&*$),&%\"zG\"\"\"F+F+\"\"#F+F+F+!\"\",(*$ )%\"xGF,F+F+*$)%\"yGF,F+F+*$)F*F,F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn := lhs(eqn) = rho^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&*$),&%\"zG\"\"\"F+F+\"\"#F+F+F+!\"\"*$)%$rhoG F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn := subs(z = rho *cos(phi) , eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&*$),&*& %$rhoG\"\"\"-%$cosG6#%$phiGF,F,F,F,\"\"#F,F,F,!\"\"*$)F+F1F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn := map(expand, eqn);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&*&)%$rhoG\"\"#\"\"\")-%$cosG 6#%$phiGF*F+F+*(F*F+F)F+F-F+F+*$F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn := map(u -> simplify(u/rho), eqn);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eqnG/*&-%$cosG6#%$phiG\"\"\",&*&%$rhoGF+F'F+F +\"\"#F+F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn := rho \+ = solve(eqn, rho);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/%$rhoG,$ *(\"\"#\"\"\"-%$cosG6#%$phiGF*,&*$)F+F)F*F*F*!\"\"F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "eqn := subs( cos(phi)^2 = 1 - sin(p hi)^2, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/%$rhoG,$*(\"\" #\"\"\"-%$cosG6#%$phiGF*-%$sinGF-!\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "testeq(rhs(eqn) = 2*cot(phi)*csc(phi) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "found := sphereplot(2*cot(phi)*csc(phi),theta=0..2*Pi,phi=Pi/ 3..Pi/2,orientation=[45,75], color = COLOR(RGB,.8,.8,.8)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(found);" }}{PARA 13 "" 1 " " {GLPLOT3D 375 375 375 {PLOTDATA 3 "6%-%%MESHG6$X.%)anythingG6\"6\"[g l'!%\"!!$^`r\":\":\"$3FF279A74590330000000000000000003FE55555555555153 FF18E48FC510C1900000000000000003FE3434C68F7C6B03FF0A87C3B2FD65C0000000 0000000003FE157E6AAFF2F8E3FEF8FB1D53E7DC300000000000000003FDF20EDCE673 6C03FEDD7FC13699A8000000000000000003FDBD52AAE35F4DE3FEC2921EE04DD5D000 00000000000003FD8C8303F9E69AE3FEA827999FCEF0300000000000000003FD5F6199 80C42EA3FE8E3639EDAC94000000000000000003FD35B653CDC0FE63FE74B49CF3902A 800000000000000003FD0F4EB3C878DC03FE5B99E5B13974200000000000000003FCD7 FA9132CDAE13FE42DDAF8AD085200000000000000003FC97326D91C29A03FE2A780213 1A22900000000000000003FC5BFB9D166AC543FE126145E9ECD2F00000000000000003 FC26145E9ECD50F3FDF5247518680D300000000000000003FBEA83DBC155D803FDC607 D9EDFE4D500000000000000003FB929FEC9AE341C3FD975F5E05531100000000000000 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