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27 " Math 2331 Fall 2002 Exa m 1" }}{PARA 0 "" 0 "" {TEXT 260 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 266 1 "1" }{TEXT 396 2 ". " }{TEXT 353 5 "If " }{XPPEDIT 394 1 "u" "I\"uG6\"" }{TEXT -1 1 " " }{TEXT 341 6 " and " }{TEXT -1 1 " \+ " }{XPPEDIT 395 1 "v" "I\"vG6\"" }{TEXT 342 19 " are vectors and " } {XPPEDIT 392 1 "u=lambda*v" "/%\"uG*&%'lambdaG\"\"\"%\"vGF&" }{TEXT 393 21 " for some scalar " }{XPPEDIT 345 1 "lambda" "I'lambdaG6\" " }{TEXT 344 31 " then which of the following " }{TEXT 346 4 "must" }{TEXT 347 11 " be true? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 5 "a) " }{XPPEDIT 381 1 "u" "I\"uG6\"" }{TEXT -1 30 " has the same direction as " }{XPPEDIT 343 1 "v" "I\"vG6\"" }{TEXT -1 11 " \nb) " }{XPPEDIT 382 1 "u" "I\"uG6\"" }{TEXT -1 21 " is parallel to " }{XPPEDIT 348 1 "v" "I\"vG6\"" }{TEXT -1 14 " \nc) " }{XPPEDIT 383 1 "u" "I\"uG6\"" }{TEXT -1 25 " \+ is perpendicular to " }{XPPEDIT 349 1 "v" "I\"vG6\"" }{TEXT -1 16 " \+ \nd) " }{XPPEDIT 384 1 "u" "I\"uG6\"" }{TEXT -1 33 " is o pposite in direction to " }{XPPEDIT 351 1 "v" "I\"vG6\"" }{TEXT -1 13 " \ne) " }{XPPEDIT 385 1 "u" "I\"uG6\"" }{TEXT -1 28 " h as the same length as " }{XPPEDIT 350 1 "v" "I\"vG6\"" }{TEXT -1 7 " \+ \nf) " }{XPPEDIT 386 1 "u" "I\"uG6\"" }{TEXT -1 32 " has magnitud e greater than " }{XPPEDIT 352 1 "v" "I\"vG6\"" }{TEXT -1 2 " " }} {PARA 258 "" 0 "" {TEXT 354 4 "g) " }{XPPEDIT 387 1 "u" "I\"uG6\"" } {TEXT -1 1 " " }{TEXT 355 27 " has magnitude less than " }{TEXT -1 1 " " }{XPPEDIT 376 1 "v" "I\"vG6\"" }{TEXT 356 18 " \nh) \+ " }{XPPEDIT 389 1 "v=(lambda)^(-1)*u" "/%\"vG*&)%'lambdaG,$\"\"\"!\" \"F(%\"uGF(" }{TEXT 388 21 " \ni) " }{XPPEDIT 391 1 "l ambda = u/v" "/%'lambdaG*&%\"uG\"\"\"%\"vG!\"\"" }{TEXT 390 8 " \nj) " }{TEXT -1 1 " " }{XPPEDIT 377 1 "u" "I\"uG6\"" }{TEXT -1 1 " " } {TEXT 378 6 " and " }{TEXT -1 1 " " }{XPPEDIT 379 1 "v" "I\"vG6\"" } {TEXT 380 89 " determine a unique plane through the origin by the for mation of linear combinations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 613 8 "Solution " }{TEXT 614 5 ": (b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "(a) is false if " }{XPPEDIT 19 1 "lambda<0" "2%'lambd aG\"\"!" }{TEXT -1 60 " for in this case the two vectors have opposit e directions." }}{PARA 0 "" 0 "" {TEXT -1 17 "(c) is false if " } {XPPEDIT 19 1 "v<>0" "0%\"vG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 19 1 "lambda<>0" "0%'lambdaG\"\"!" }{TEXT -1 38 " for in this case the dot product of " }{XPPEDIT 19 1 "u" "I\"uG6\"" }{TEXT -1 7 " and " } {XPPEDIT 19 1 "v" "I\"vG6\"" }{TEXT -1 21 " is the product of " } {XPPEDIT 19 1 "lambda" "I'lambdaG6\"" }{TEXT -1 24 " and the magnitud e of " }{XPPEDIT 19 1 "v" "I\"vG6\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 17 "(d) is false if " }{XPPEDIT 19 1 "lambda>0" "2\"\"!%'l ambdaG" }{TEXT -1 59 " for in this case the two vectors have the same direction." }}{PARA 0 "" 0 "" {TEXT -1 18 "(e) is false if " } {XPPEDIT 19 1 "abs(lambda)<>1" "0-%$absG6#%'lambdaG\"\"\"" }{TEXT -1 25 " for the magnitude of " }{XPPEDIT 19 1 "u" "I\"uG6\"" }{TEXT -1 5 " is " }{XPPEDIT 19 1 "abs(lambda)" "-%$absG6#%'lambdaG" }{TEXT -1 27 " times the magnitude of " }{XPPEDIT 19 1 "v" "I\"vG6\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "(f) is false if " } {XPPEDIT 19 1 "abs(lambda)<=1" "1-%$absG6#%'lambdaG\"\"\"" }{TEXT -1 25 " for the magnitude of " }{XPPEDIT 19 1 "u" "I\"uG6\"" }{TEXT -1 5 " is " }{XPPEDIT 19 1 "abs(lambda)" "-%$absG6#%'lambdaG" }{TEXT -1 27 " times the magnitude of " }{XPPEDIT 19 1 "v" "I\"vG6\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "(g) is false if " } {XPPEDIT 19 1 "abs(lambda)>=1" "1\"\"\"-%$absG6#%'lambdaG" }{TEXT -1 25 " for the magnitude of " }{XPPEDIT 19 1 "u" "I\"uG6\"" }{TEXT -1 5 " is " }{XPPEDIT 19 1 "abs(lambda)" "-%$absG6#%'lambdaG" }{TEXT -1 27 " times the magnitude of " }{XPPEDIT 19 1 "v" "I\"vG6\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "(h) is false if " } {XPPEDIT 19 1 "u" "I\"uG6\"" }{TEXT -1 25 " is the zero vector and " }{XPPEDIT 19 1 "lambda = 0" "/%'lambdaG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 91 "(i) is false (and does not even make sense) bec ause the quotient of vectors is not defined." }}{PARA 0 "" 0 "" {TEXT -1 72 "(j) is false because (b) is true! Algebraically, the linear com bination " }{XPPEDIT 19 1 "su+tv" ",&%#suG\"\"\"%#tvGF$" }{TEXT -1 18 " can bewritten as " }{XPPEDIT 19 1 "(s*lambda+t)*v" "*&,&*&%\"sG\"\" \"%'lambdaGF&F&%\"tGF&F&%\"vGF&" }{TEXT -1 57 "; therefore, the set o f all linear combinations of " }{XPPEDIT 19 1 "u" "I\"uG6\"" } {TEXT -1 7 " and " }{XPPEDIT 19 1 "v" "I\"vG6\"" }{TEXT -1 31 ". is \+ a line through the origin." }}{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 263 "" 0 "" {TEXT 299 4 "2. " } {TEXT -1 29 "If the magnitude of vector " }{XPPEDIT 19 1 "v=`<`*2,b, -4*`>`" "6%/%\"vG*&%\"GF'!\"\"" } {TEXT -1 20 " is 6 and if " }{XPPEDIT 19 1 "v " "I\"vG6\"" } {TEXT -1 36 " has the same direction \n as " }{XPPEDIT 19 1 "`< `*a,1, c*`>`" "6%*&%\"GF%" }{TEXT -1 18 " then what is " }{XPPEDIT 19 1 "b" "I\"bG6\"" }{TEXT -1 2 "? " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 357 4 "a) " } {XPPEDIT 358 1 "0" "\"\"!" }{TEXT 359 9 " b) " }{XPPEDIT 360 1 "1 " "\"\"\"" }{TEXT 361 10 " c) " }{XPPEDIT 362 1 "-1" ",$\"\"\"! \"\"" }{TEXT 363 7 " d) " }{XPPEDIT 364 1 "2" "\"\"#" }{TEXT 365 10 " e) " }{XPPEDIT 366 1 "-2" ",$\"\"#!\"\"" }{TEXT 367 11 " \+ f) " }{XPPEDIT 368 1 "3" "\"\"$" }{TEXT 369 11 " g) " } {XPPEDIT 370 1 "-3" ",$\"\"$!\"\"" }{TEXT 371 8 " h) " }{XPPEDIT 372 1 "4" "\"\"%" }{TEXT 373 8 " i) " }{XPPEDIT 374 1 "-4" ",$\"\" %!\"\"" }{TEXT 375 76 " \nj) Insufficient inf ormation to determine answer " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 615 8 "Solution" }{TEXT 618 5 ": (h)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "D efine a length function:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "vectorLength := v -> sqrt(v[1]^2 + \+ v[2]^2 + v[3]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-vectorLengthG: 6#%\"vG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,(*$&9$6#\"\"\"\"\"#F4*$&F2 6#F5F5F4*$&F26#\"\"$F5F4F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "The magnitude of " }{XPPEDIT 19 1 "v=`< `*2,b,-4*`>`" "6%/%\"vG*&%\"GF'!\" \"" }{TEXT -1 4 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "vectorLength([2, b, -4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&\"#?\"\"\"*$%\"bG\"\"#F&#F&F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Therefore b satisf ies" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn_for_b := (20+b^2)^(1/2) = 6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*eqn_for_bG/*$,&\"#?\"\"\"* $%\"bG\"\"#F)#F)F,\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 9 "We solve:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "so lve( eqn_for_b , b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%!\"%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 616 2 "If" } {TEXT -1 102 " we had no further information then we would not know wh ether b = 4 or b = - 4. The correct answer " }{TEXT 617 5 "would" } {TEXT -1 14 " then be (j) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "But we are told that " }{XPPEDIT 19 1 "v=`<`*2, b,-4*`>`" "6%/%\"vG*&%\"GF'!\"\"" } {TEXT -1 30 " has the same direction as " }{XPPEDIT 19 1 "`<`*a,1, \+ c*`>`" "6%*&%\"GF%" }{TEXT -1 119 ". Th erefore the sign of b is the same as the sign of 1, namely posi tive. So we can say for sure that b = 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 263 18 "3. If the vectors " } {TEXT 400 2 " " }{XPPEDIT 403 1 "`<`*3,-b,b*`>`" "6%*&%\"GF%" }{TEXT 397 1 " " }{TEXT 401 3 "and" } {TEXT 402 2 " " }{XPPEDIT 404 1 "`<`*2,-1,-2*b*`>`" "6%*&%\"GF%F(" }{TEXT 398 1 " " }{TEXT 405 34 " are perpendicular then what might" }{TEXT 406 1 " " }{XPPEDIT 407 1 "b" "I\"bG6\"" }{TEXT 399 2 " " }{TEXT 408 4 "be? " }{TEXT 409 2 " " }}{PARA 3 "" 0 "" {TEXT 298 154 "a) -1 b) -1/2 \+ c) 0 d) 1/2 e) 1 \nf) 3/2 g) 2 h) 5/2 i) 3 j) 7/2 " }{TEXT 410 7 " \n" }}{PARA 0 "" 0 "" {TEXT 619 8 "Solution" }{TEXT 620 5 ": (g )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We n eed the dot product - it is in the " }{TEXT 621 6 "linalg" }{TEXT -1 10 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eqn_for_b := dotprod([3, -b, b],[2, -1, -2*b]) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*eqn_for_bG/,(\"\"'\"\"\"%\"bGF(*$ F)\"\"#!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(e qn_for_b, b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\"$\"\"#F%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "So " } {XPPEDIT 657 1 "b" "I\"bG6\"" }{TEXT 656 1 " " }{TEXT -1 75 "might be \+ one of these two numbers, of which only one is in the answer list." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 1 " " }{TEXT 417 36 "4. What is the angle between vectors" }{TEXT 418 3 " " }{XPPEDIT 415 1 "`<`*1,2,2*`>`" "6%*&%\"GF%" }{TEXT 411 1 " " }{TEXT 413 3 "and" }{TEXT 414 2 " " } {XPPEDIT 416 1 "`<`*(3-2*sqrt(11)),6+sqrt(11),0*`>`" "6%*&%\"GF%" }{TEXT 412 3 " ?" }}{PARA 0 "" 0 "" {TEXT 270 4 "a) " } {XPPEDIT 256 0 "Pi/6" "*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 257 11 " \+ " }{TEXT 419 3 "b) " }{TEXT 420 1 " " }{XPPEDIT 304 0 "Pi/4" "*&%# PiG\"\"\"\"\"%!\"\"" }{TEXT 305 14 " c) " }{XPPEDIT 306 0 "P i/3" "*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 307 7 " " }{TEXT 421 5 " \+ d) " }{XPPEDIT 308 0 "Pi/2" "*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 309 5 " \+ " }{TEXT 422 30 " e)" }{TEXT 423 3 " \+ " }{XPPEDIT 312 0 "2*Pi/3" "*(\"\"#\"\"\"%#PiGF$\"\"$!\"\"" }{TEXT 313 4 " " }}{PARA 262 "" 0 "" {TEXT 261 4 "f) " }{TEXT 258 2 " " }{XPPEDIT 310 0 "3*Pi/4" "*(\"\"$\"\"\"%#PiGF$\"\"%!\"\"" }{TEXT 311 13 " g) " }{XPPEDIT 314 0 "5*Pi/6" "*(\"\"&\"\"\"%#PiGF$\"\"' !\"\"" }{TEXT 315 16 " h) " }{XPPEDIT 316 0 "Pi" "I#PiG%*p rotectedG" }{TEXT 317 15 " i) " }{XPPEDIT 318 0 "arccos((3+ sqrt(11))/10)" "-%'arccosG6#*&,&\"\"$\"\"\"-%%sqrtG6#\"#6F(F(\"#5!\"\" " }{TEXT 259 13 " j) " }{XPPEDIT 320 0 "0" "\"\"!" }{TEXT 321 1 " " }{TEXT 319 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 622 8 "Solution" }{TEXT 623 6 ": (c)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "v := vector([1,2,2]); w := vector([3-2*sq rt(11),6+sqrt(11),0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VEC TORG6#7%\"\"\"\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECT ORG6#7%,&\"\"$\"\"\"*$\"#6#F+\"\"#!\"#,&\"\"'F+F,F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eqn := dotprod(v,w) = vectorLength( v)*vectorLength(w)*cos(theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e qnG/\"#:,$*&,&*$,&\"\"$\"\"\"*$\"#6#F-\"\"#!\"#F1F-*$,&\"\"'F-F.F-F1F- F0-%$cosG6#%&thetaGF-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " soln := theta = solve(eqn, theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%solnG/%&thetaG-%'arccosG6#,$*$,&*$,&\"\"$\"\"\"*$\"#6#F0\"\"#!\"#F4 F0*$,&\"\"'F0F1F0F4F0#!\"\"F4\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% &thetaG,$%#PiG#\"\"\"\"\"$" }}}{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 424 34 "What is the radius of the sphere " } {TEXT 427 1 " " }{XPPEDIT 425 1 "x^2+y^2+z^2-10*x+6*y-4*z+2=0" "/,0*$% \"xG\"\"#\"\"\"*$%\"yGF&F'*$%\"zGF&F'*&\"#5F'F%F'!\"\"*&\"\"'F'F)F'F'* &\"\"%F'F+F'F.F&F'\"\"!" }{TEXT 426 2 " ?" }}{PARA 261 "" 0 "" {TEXT -1 50 "a) 1 b) 2 c) 3 d) 4 e) 5" }}{PARA 265 "" 0 "" {TEXT -1 53 "f) 6 g) 7 h) 8 i) 9 \+ j) 10" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 624 8 "Solution" }{TEXT 625 6 ": (f)\n" }}{PARA 0 "" 0 "" {TEXT -1 52 "The re is a command for completing the square in the " }{TEXT 626 7 "stude nt" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "The next line applies (o r \"maps\") the command \"complete the square of the argument with res pect to x\"to both sides of the equation that is given as the second a rgument:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "map(u -> compl etesquare(u,x), x^2+y^2+z^2-10*x+6*y-4*z+2 = 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.*$,&%\"xG\"\"\"!\"&F(\"\"#F(!#BF(*$%\"yGF*F(*$%\"zGF *F(F-\"\"'F/!\"%\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ma p(u -> completesquare(u,y), \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,,*$,&%\"yG\"\"\"\"\"$F(\"\"#F(!#KF(*$,&%\"xGF(!\"&F(F*F(*$%\"zGF*F(F1 !\"%\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "map(u -> compl etesquare(u,z), \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$,&%\"zG \"\"\"!\"#F(\"\"#F(!#OF(*$,&%\"yGF(\"\"$F(F*F(*$,&%\"xGF(!\"&F(F*F(\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "map(u -> u+36, \"); \+ # Add 36 to both sides of equation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,(*$,&%\"zG\"\"\"!\"#F(\"\"#F(*$,&%\"yGF(\"\"$F(F*F(*$,&%\"xGF(!\"&F (F*F(\"#O" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 267 6 "6. If \+ " }{TEXT 436 2 " " }{XPPEDIT 262 1 "`<`*a,b,c*`>`" "6%*&%\"GF%" }{TEXT 428 1 " " }{TEXT 437 29 " is the v ector projection of " }{TEXT 438 2 " " }{XPPEDIT 429 1 "`<`*3,4,-4*`> `" "6%*&%\"GF%!\"\"" }{TEXT 430 2 " \+ " }{TEXT 439 3 " on" }{TEXT 440 3 " " }{XPPEDIT 431 1 "`<`*2,1,-2*`> `" "6%*&%\"GF%!\"\"" }{TEXT 432 1 " " } {TEXT 441 13 " then what is" }{TEXT 442 1 " " }{TEXT 434 1 " " } {XPPEDIT 435 0 "a" "I\"aG6\"" }{TEXT 433 2 " ?" }}{PARA 0 "" 0 "" {TEXT 268 128 "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 627 8 " Solution" }{TEXT 628 6 ": (j)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "v := vector([3,4,-4]); w := vector([2,1,-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"$\"\"%!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7%\"\"#\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "k := dotprod(v,w)/dotprod(w,w);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "scalarmul(w,k); # Multiplies vector w by scalar \+ k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"%\"\"#!\"%" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 269 8 "7. Let " }{TEXT 468 1 " " }{XPPEDIT 443 1 "P" "I\"PG6\"" }{TEXT 444 1 " " }{TEXT 452 31 " be the projection (vector) of " }{TEXT 453 2 " " }{XPPEDIT 445 1 "v=`<`*6,-1,8*`>`" "6%/%\"vG*&%\"GF'" }{TEXT 446 2 " " }{TEXT 454 3 " on" }{TEXT 455 3 " " } {XPPEDIT 447 1 "`<`*1,2,3*`>`" "6%*&%\"G F%" }{TEXT 448 1 " " }{TEXT 456 7 ". Let " }{TEXT 459 1 " " } {XPPEDIT 466 1 "Q=`<`*a,b,c*`>`" "6%/%\"QG*&%\"GF'" }{TEXT 467 1 " " }{TEXT 460 39 " be the vector that is \+ perpendicular to" }{TEXT 469 2 " " }{XPPEDIT 461 1 "P" "I\"PG6\"" } {TEXT 462 16 " and such that " }{TEXT 463 1 " " }{XPPEDIT 464 1 "v=P+ Q" "/%\"vG,&%\"PG\"\"\"%\"QGF&" }{TEXT 465 10 ". What is" }{TEXT 457 1 " " }{TEXT 450 1 " " }{XPPEDIT 451 0 "b" "I\"bG6\"" }{TEXT 449 2 " ? " }{TEXT 458 1 "\n" }}{PARA 0 "" 0 "" {TEXT 470 128 "a) -5 b) \+ -4 c) -3 d) -2 e) -1 \n\n f) 0 g) 1 \+ h) 2 i) 3 j) 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 629 8 "Solution" }{TEXT 630 5 ": (a)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "v := vector([6,-1,8]); w := vector([1,2,3]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"'!\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "k := dotprod(v,w)/dotprod(w,w);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P := scalarmul(w,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'VECTORG6#7%\"\"#\"\"%\"\"'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "Q := evalm(v-P); # v-P is a matrix: evalm evaluates it." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'VECTORG6#7%\"\"%!\"& \"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 488 9 "8. Let " } {XPPEDIT 480 1 "`<`*a,b,c*`>`" "6%*&%\"GF%" }{TEXT 481 1 " " }{TEXT 476 28 " be the vector cross product" }{TEXT 486 1 " " }{XPPEDIT 484 1 "`<`*2,1,1*`>` " "6%*&%\"GF%" }{TEXT 485 1 " " }{TEXT 487 2 " " }{TEXT 482 1 "X " }{TEXT 483 1 " " }{TEXT 477 1 " " }{XPPEDIT 471 1 "`<`*(-4),3,1*`>` " "6%*&%\"GF%" }{TEXT 472 1 " " } {TEXT 478 9 ". What is" }{TEXT 479 1 " " }{TEXT 474 1 " " }{XPPEDIT 475 0 "a+b+c" ",(%\"aG\"\"\"%\"bGF$%\"cGF$" }{TEXT 473 2 " ?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 489 128 "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 631 8 "Solution" }{TEXT 632 5 ": (h)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "v := vecto r([2,1,1]); w := vector([-4,3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"vG-%'VECTORG6#7%\"\"#\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"wG-%'VECTORG6#7%!\"%\"\"$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cp := crossprod(v,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%!\"#!\"'\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cp[1]+cp[2]+cp[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 54 "9. What is the sine of the angle between the vectors " }{XPPEDIT 490 1 "`<`*2,2,1*`>` " "6%*&% \"GF%" }{TEXT 491 7 " and " }{XPPEDIT 492 1 "`<`*3,0,4*`>` " "6%*&%\"GF%" } {TEXT 493 2 " ?" }{TEXT 494 1 " " }}{PARA 258 "" 0 "" {TEXT 495 5 "\na ) " }{XPPEDIT 496 1 "1/6" "*&\"\"\"F#\"\"'!\"\"" }{TEXT 497 9 " b ) " }{XPPEDIT 498 1 "1/3" "*&\"\"\"F#\"\"$!\"\"" }{TEXT 499 10 " \+ c) " }{XPPEDIT 500 1 "2/3" "*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 501 7 " \+ d) " }{XPPEDIT 502 1 "sqrt(5)/15" "*&-%%sqrtG6#\"\"&\"\"\"\"#:!\"\" " }{TEXT 503 10 " e) " }{XPPEDIT 504 1 "sqrt(5)/5" "*&-%%sqrtG6# \"\"&\"\"\"F&!\"\"" }{TEXT 505 9 " f) " }{XPPEDIT 506 1 "sqrt(5)/ 3" "*&-%%sqrtG6#\"\"&\"\"\"\"\"$!\"\"" }{TEXT 507 11 " g) " } {XPPEDIT 508 1 "sqrt(3)/2" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT 509 8 " h) " }{XPPEDIT 510 1 "sqrt(3)/3" "*&-%%sqrtG6#\"\"$\"\"\"F &!\"\"" }{TEXT 511 8 " i) " }{XPPEDIT 512 1 "sqrt(3)/4" "*&-%%sqrt G6#\"\"$\"\"\"\"\"%!\"\"" }{TEXT 513 76 " \nj ) Insufficient information to determine answer " }{TEXT -1 1 " " }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 633 8 "Solution " }{TEXT 634 6 ": (f)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "v := vector([2,2,1]); w := vector( [3,0,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"# F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7%\"\"$ \"\"!\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cp := crosspr od(v,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%\"\")! \"&!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "eqn := vectorLen gth(cp) = vectorLength(v)*vectorLength(w)*sin(theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*$\"\"&#\"\"\"\"\"#F(,$-%$sinG6#%&thetaG \"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sin(theta) = solve( eqn, sin(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%&theta G,$*$\"\"&#\"\"\"\"\"##F,\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "eqn1 := dotprod(v,w) = vectorLength(v)*vectorLength(w)*cos(theta); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/\"#5,$-%$cosG6#%&thetaG\" #:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eqn2 := cos(theta) = \+ solve(eqn1, cos(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/- %$cosG6#%&thetaG#\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sin(theta) = subs(eqn2, sqrt(1-cos(theta)^2));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$sinG6#%&thetaG,$*&\"\"&#\"\"\"\"\"#\"\"*F+#F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG,$*$\"\"&#\"\"\"\"\"##F,\" \"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 273 39 "10. What is t he area of parallelogram " }{TEXT 514 4 "PQRS" }{TEXT 515 5 " if " } {TEXT 516 1 "P" }{TEXT 517 12 " = (4,3,5), " }{TEXT 518 1 "Q" }{TEXT 519 16 " = (1,2,3), and " }{TEXT 520 1 "R" }{TEXT 521 11 " = (3,4,6)? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 21 "a) \+ 4 b) " }{XPPEDIT 323 0 "3*sqrt(2)" "*&\"\"$\"\"\"-%%sqrtG6 #\"\"#F$" }{TEXT 324 16 " c) " }{XPPEDIT 325 0 "2*sqrt(6) " "*&\"\"#\"\"\"-%%sqrtG6#\"\"'F$" }{TEXT 326 19 " d) \+ " }{XPPEDIT 327 0 "5" "\"\"&" }{TEXT 328 21 " e) " } {XPPEDIT 329 0 "4*sqrt(2)" "*&\"\"%\"\"\"-%%sqrtG6#\"\"#F$" }{TEXT 330 21 " \nf) " }{XPPEDIT 331 0 "6" "\"\"'" }{TEXT 332 14 " g) " }{XPPEDIT 333 0 "sqrt(42)" "-%%sqrtG6#\"#U" } {TEXT 334 18 " h) " }{XPPEDIT 335 0 "4*sqrt(3)" "*&\"\"% \"\"\"-%%sqrtG6#\"\"$F$" }{TEXT 336 19 " i) " } {XPPEDIT 337 0 "2*sqrt(13)" "*&\"\"#\"\"\"-%%sqrtG6#\"#8F$" }{TEXT 338 16 " j) " }{XPPEDIT 339 0 "sqrt(58)" "-%%sqrtG6#\"#e" }}{PARA 0 "" 0 "" {TEXT 340 1 " " }}{PARA 0 "" 0 "" {TEXT 635 8 "Solut ion" }{TEXT 636 6 ": (g)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "OP := vector([4,3,5]); \nOQ := vector([1,2,3]); \nOR := vector([3 ,4,6]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OPG-%'VECTORG6#7%\"\"% \"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OQG-%'VECTORG6#7%\"\" \"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ORG-%'VECTORG6#7%\" \"$\"\"%\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v := evalm (OP-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"$ \"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w := evalm(O R-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7%\"\"#F) \"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cp := crossprod(v, w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%!\"\"!\"&\" \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vectorLength(cp);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"#U#\"\"\"\"\"#" }}}{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 form ed from the three vectors" }{TEXT 528 1 " " }{XPPEDIT 522 1 "`<`*2,2,1 *`>` " "6%*&%\"GF%" }{TEXT 523 3 ", " } {XPPEDIT 524 1 "`<`*1,0,1*`>` " "6%*&%\"GF% " }{TEXT 525 3 ", " }{TEXT 530 3 "and" }{TEXT 531 2 " " }{XPPEDIT 526 1 "`<`*1,-1,4*`>` " "6%*&%\"GF% " }{TEXT 527 1 "?" }{TEXT 529 3 " \n\n" }{TEXT 303 70 "a) 1 \+ b) 2 c) 3 d) 4 e) 5" }{TEXT 532 21 " " }}{PARA 0 "" 0 "" {TEXT 534 34 "f) 6 \+ g) 7 " }{TEXT -1 3 " h)" }{TEXT 533 36 " 8 \+ i) 9 j) 12" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 637 8 "Solution" }{TEXT 638 6 ": (e)\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "u := vector([2,2,1]);\nv := vector([1,0,1]);\nw := vector([1,-1,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'VECTORG6#7%\"\"#F)\"\"\"" }}{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 31 "abs(dotprod(u,crossprod(v,w)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 275 83 "12. The line through the points ( 3, -1, 2) and (5, 2 , -1) passes through a point (" }{TEXT 541 1 "a" }{TEXT 542 2 ", " } {TEXT 543 1 "b" }{TEXT 544 15 ", 8). What is " }{TEXT 539 1 "a" } {TEXT 540 79 " ? \n\na) -4 b) -3 c) -2 \+ d) -1 e) 0" }{TEXT 536 21 " " }} {PARA 0 "" 0 "" {TEXT 537 36 "f) 1 g) 2 " } {TEXT -1 1 " " }{TEXT 538 39 "h) 3 i) 4 j) \+ " }{TEXT -1 1 " " }{TEXT 545 1 "5" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT 535 1 " " }}{PARA 0 "" 0 "" {TEXT 639 8 "Solution" }{TEXT 640 6 ": (d)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "OP := vector([3, -1,2]); \nOQ := vector([5,2,-1]); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#OPG-%'VECTORG6#7%\"\"$!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OQG-%'VECTORG6#7%\"\"&\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v := evalm(OQ-OP);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"vG-%'VECTORG6#7%\"\"#\"\"$!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "line := evalm(OP + scalarmul(v,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%lineG-%'VECTORG6#7%,&\"\"$\"\"\"%\"tG\"\"#,&!\"\" F+F,F*,&F-F+F,!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn : = t = solve(line[3] = 8, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqn G/%\"tG!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a = subs(eqn , line[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG!\"\"" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 276 1 " " } {TEXT 550 17 "13. The vector " }{TEXT 546 1 "<" }{TEXT 547 12 " a , \+ b , -2 " }{TEXT 548 1 ">" }{TEXT 549 69 " is perpendicular to the pla ne z = 2x - y + 1. What is a ? " }{TEXT 551 3 " \n\n" }{TEXT 552 132 "a) -4 b) - 3 c) - 2 d) - 1 e) 0 \nf) 1 g) 2 h) 3 i) 4 j) \+ 5" }}{PARA 0 "" 0 "" {TEXT 641 9 "\nSolution" }{TEXT 642 6 ": (i)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "2*x-y-z = -1; # Standard form for plane" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"xG\"\"#%\"yG!\"\"%\"zGF(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 63 "eqn1 := vector([a,b,-2]) = scalarmul(vector([2 ,-1,-1]),lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%'VECTO RG6#7%%\"aG%\"bG!\"#-F'6#7%,$%'lambdaG\"\"#,$F1!\"\"F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn2 := map( z -> z[3], eqn1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/!\"#,$%'lambdaG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn3 := lambda = solve(eqn2 \+ , lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/%'lambdaG\"\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a = subs(eqn3, rhs(eqn1 )[1]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 266 "" 0 "" {TEXT -1 84 "14. A plane passes through the three points \+ (1, 0, 0), (0, 2, 0), and (2, 3, 2). " }}{PARA 3 "" 0 "" {TEXT 553 51 "Its equation is Ax + 2y + Cz = D. What is D? " }{TEXT 556 1 "\n" }}{PARA 3 "" 0 "" {TEXT 555 132 "a) -4 b) - 3 c) - 2 d) - 1 e) 0 \nf) 1 g) 2 h) \+ 3 i) 4 j) 5" }{TEXT 554 1 "\n" }}{PARA 0 "" 0 "" {TEXT 643 8 "Solution" }{TEXT 644 6 ": (i)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "OP := vector([1,0,0]); \nOQ := vector([0,2,0]); \+ \nOR := vector([2,3,2]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OPG-%' VECTORG6#7%\"\"\"\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OQG-%'V ECTORG6#7%\"\"!\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ORG-%'VEC TORG6#7%\"\"#\"\"$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v : = evalm(OP-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7 %\"\"\"!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w := ev alm(OR-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7%\" \"#\"\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cp := crossp rod(v,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%!\"%! \"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "eqn1 := cp[1]*( x-1)+cp[2]*(y-0)+cp[3]*(z-0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% eqn1G/,*%\"xG!\"%\"\"%\"\"\"%\"yG!\"#%\"zG\"\"&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn2 := map(u->-u, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,*%\"xG\"\"%!\"%\"\"\"%\"yG\"\"#%\"zG!\"& \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn3 := map(u->u+4 , eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,(%\"xG\"\"%%\"yG \"\"#%\"zG!\"&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "D = rhs (eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"DG\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 559 164 "15. A pla ne is perpendicular to the line (x - 1)/3 = 1 - y = 2z . The plane passes through the point (1, 2, 2) as well as the point ( 0, - 2, c ) . What is c ?" }{TEXT 560 3 " \n" }{TEXT 558 133 "\na) -4 \+ b) - 3 c) - 2 d) - 1 e) 0 \nf) 1 g) 2 h) 3 i) 4 j) 5" }{TEXT 557 1 "\n" }} {PARA 0 "" 0 "" {TEXT 652 8 "Solution" }{TEXT 653 6 ": (a)\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "normalVector := vector([3,-1 ,1/2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-normalVectorG-%'VECTORG6 #7%\"\"$!\"\"#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "OP := vector([1,2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OPG-% 'VECTORG6#7%\"\"\"\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "eqn_for_plane := normalVector[1]*(x-OP[1])\n + \+ normalVector[2]*(y-OP[2])\n + normalVector[3]*(z- OP[3]) = 0;\n\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.eqn_for_planeG/ ,*%\"xG\"\"$!\"#\"\"\"%\"yG!\"\"%\"zG#F*\"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn_for_c := subs(\{x=0,y=-4,z=c\}, eqn_for _plane); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*eqn_for_cG/,&\"\"#\"\" \"%\"cG#F(F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "c = so lve(eqn_for_c, c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG!\"%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 265 111 "16. \+ The two planes x + y + z = 6 and - 4x - y + 2z = 0 intersect in a line that is parallel to the \nvector" }{TEXT 568 2 " " }{TEXT 563 1 "<" }{TEXT 564 11 " 1 , b , c " }{TEXT 565 1 ">" }{TEXT 566 16 ". \+ What is b ? " }{TEXT 567 3 " \n\n" }{TEXT 562 132 "a) -4 b) - 3 c) - 2 d) - 1 e) 0 \nf) 1 g) 2 h) 3 i) 4 j) 5" }{TEXT 561 1 "\n" }} {PARA 0 "" 0 "" {TEXT 647 8 "Solution" }{TEXT 648 6 ": (c)\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn1 := x + y + z = 6;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,(%\"xG\"\"\"%\"yGF(%\"zGF(\" \"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn2 := -4*x - y + 2 *z = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,(%\"xG!\"%%\"yG! \"\"%\"zG\"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn3 := subs(x=t, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,(%\" tG\"\"\"%\"yGF(%\"zGF(\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn4 := subs(x=t, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn 4G/,(%\"tG!\"%%\"yG!\"\"%\"zG\"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(\{eqn3,eqn4\}, \{y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"zG,&%\"tG\"\"\"\"\"#F(/%\"yG,&F'!\"#\"\"%F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "parEqns := x = t, y = -2*t + 4, z = t + 2; # Parametric eqns" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% (parEqnsG6%/%\"xG%\"tG/%\"yG,&F(!\"#\"\"%\"\"\"/%\"zG,&F(F.\"\"#F." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "b = coeff(rhs(parEqns[2]),t );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "More simply (using vector methods in a somewhat more sophistica ted manner):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "N1 := [1,1,1]:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "N2 := [-4,-1,2]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "L := linalg[crossprod](N1,N2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG-%'VECTORG6#7%\"\"$!\"'F) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "map(z->z/3, L); #to ge t 1 in first entry" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\" \"\"!\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# so, b = -2 \+ and c = 1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 267 " " 0 "" {TEXT -1 163 "17. A plane is parallel to the plane x + 2y + 3z \+ = 0 and also passes through the point (1, 1, 1). What is the sum of it s three intercepts with the coordinate axes?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 149 "a) 4 b) 5 \+ c) 6 d) 7 e) 8 \nf) 9 \+ g) 10 h) 11 i) 12 j) 13" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 645 8 "Solution" }{TEXT 646 6 ": (h)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn := 1*(x-1)+2 *(y-1)+3*(z-1) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,*%\"xG \"\"\"!\"'F(%\"yG\"\"#%\"zG\"\"$\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x_intercept := solve(subs(\{y=0,z=0\}, eqn),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%,x_interceptG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "y_intercept := solve(subs(\{x=0,z=0\}, eq n),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,y_interceptG\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "z_intercept := solve(subs(\{ x=0,y=0\}, eqn),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,z_interceptG \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "x_intercept + y_in tercept + z_intercept;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 268 "" 0 "" {TEXT -1 136 "18. Examine the equations in \+ each of answers (a) through (h). Determine which, if any, of these eq uations have graphs that consist of " }{TEXT 587 3 "two" }{TEXT -1 55 " sheets separated by a positive distance. If there is " }{TEXT 300 11 "exactly one" }{TEXT 301 1 " " }{TEXT -1 116 "such equation, choose that as your answer. Otherwise, choose (i) or (j), whichever is appro priate, as your answer. \n" }}{PARA 269 "" 0 "" {TEXT -1 3 "a) " } {TEXT 571 1 " " }{XPPEDIT 572 1 "x^2+4*x+y^2+z^2-2*z=0" "/,,*$%\"xG\" \"#\"\"\"*&\"\"%F'F%F'F'*$%\"yGF&F'*$%\"zGF&F'*&F&F'F-F'!\"\"\"\"!" } {TEXT 570 2 " " }{TEXT -1 18 " b) " }{TEXT 574 1 " " } {XPPEDIT 575 1 "x^2+ y^2 - z^2 =0" "/,(*$%\"xG\"\"#\"\"\"*$%\"yGF&F'* $%\"zGF&!\"\"\"\"!" }{TEXT -1 5 " \nc) " }{TEXT 576 1 " " }{XPPEDIT 577 1 "x^2+ y^2 - z^2 = 1" "/,(*$%\"xG\"\"#\"\"\"*$%\"yGF&F'*$%\"zGF& !\"\"F'" }{TEXT -1 41 " d) " } {XPPEDIT 578 1 "x^2 - y^2 - z^2 = 1" "/,(*$%\"xG\"\"#\"\"\"*$%\"yGF&! \"\"*$%\"zGF&F*F'" }{TEXT -1 7 " \ne) " }{XPPEDIT 580 1 "x^2+ y^2 - z= 1" "/,(*$%\"xG\"\"#\"\"\"*$%\"yGF&F'%\"zG!\"\"F'" }{TEXT -1 43 " \+ f) " }{XPPEDIT 582 1 "x^2+ y^2 \+ - z = 0" "/,(*$%\"xG\"\"#\"\"\"*$%\"yGF&F'%\"zG!\"\"\"\"!" }{TEXT 583 1 "\n" }{TEXT -1 5 "g) " }{XPPEDIT 584 1 "x^2 - y^2 - z= 0" "/, (*$%\"xG\"\"#\"\"\"*$%\"yGF&!\"\"%\"zGF*\"\"!" }{TEXT -1 42 " \+ h) " }{XPPEDIT 585 1 "x^2 - y^2 + z= 0 " "/,(*$%\"xG\"\"#\"\"\"*$%\"yGF&!\"\"%\"zGF'\"\"!" }{TEXT -1 3 " " }{TEXT 302 1 "\n" }{TEXT -1 74 "i) not one of the equations above has a graph that consists of two sheets" }}{PARA 270 "" 0 "" {TEXT -1 79 "j) more than one of the equations above have graphs that consist of \+ two sheets" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 650 8 "Solution" }{TEXT 651 5 ": (d)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "(a) - sphere (one sheet)" }}{PARA 0 "" 0 "" {TEXT -1 61 "(b) - two cones attached at origin (perhaps two shee ts, but " }{TEXT 649 3 "not" }{TEXT -1 35 " separated by a positive \+ distance)" }}{PARA 0 "" 0 "" {TEXT -1 94 "(c) - hyperboloid of one she et (horizontal traces are circles, vertical traces are hyperbolas)" }} {PARA 0 "" 0 "" {TEXT -1 117 "(d) - hyperboloid of two sheets (horizon tal traces are hyperbolas, vertical traces in x = k are circles for k^ 2 >= 1)" }}{PARA 0 "" 0 "" {TEXT -1 42 "(e) - paraboloid of revolution (one sheet)" }}{PARA 0 "" 0 "" {TEXT -1 42 "(f) - paraboloid of revol ution (one sheet)" }}{PARA 0 "" 0 "" {TEXT -1 24 "(g) - saddle (one sh eet)" }}{PARA 0 "" 0 "" {TEXT -1 24 "(h) - saddle (one sheet)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 610 187 "19. At least one of the following equations, some us ing cylindrical equations, some using spherical coordinates, is the eq uation of a sphere. Examine choices (a) through (h). If there is " } {TEXT 608 11 "exactly one" }{TEXT 609 1 " " }{TEXT 611 125 "such equat ion among them, choose that as your answer. Otherwise, choose (i) or ( j), whichever is appropriate, as your answer." }}{PARA 3 "" 0 "" {TEXT 586 4 "a) " }{XPPEDIT 588 0 "rho=sin(phi)" "/%$rhoG-%$sinG6#%$p hiG" }{TEXT 589 20 " b) " }{XPPEDIT 590 0 "r=2" "/%\"r G\"\"#" }{TEXT 591 18 " \nc) " }{XPPEDIT 592 0 "phi=Pi/3 " "/%$phiG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 593 28 " \+ d) " }{XPPEDIT 594 0 "r= cos(theta)" "/%\"rG-%$cosG6#%&thetaG" } {TEXT 595 17 " \ne) " }{XPPEDIT 596 0 "rho=cos(theta)" "/% $rhoG-%$cosG6#%&thetaG" }{TEXT 597 21 " f) " } {XPPEDIT 598 0 "rho=2" "/%$rhoG\"\"#" }{TEXT 599 16 " \ng) \+ " }{XPPEDIT 600 0 "r^2+z^2=1" "/,&*$%\"rG\"\"#\"\"\"*$%\"zGF&F'F'" } {TEXT 601 21 " h) " }{XPPEDIT 602 0 "theta=Pi/2" "/%& thetaG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 603 1 " " }{TEXT 604 2 " " } {TEXT 606 26 "(cylindrical coordinates) " }{TEXT 607 1 " " }{TEXT 605 54 " \ni) two spheres above j) three spheres above" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "plot3d(sin(phi), theta=0..2*Pi, ph i=0..Pi, coords=spherical,scaling=constrained); # (a) Donut, No spher e" }}{PARA 13 "" 1 "" {INLPLOT "6$-%%MESHG6#7;7;7%\"\"!F)F)7%$\"1#eYbo 3Pq\"!#A *=q7IVF07%$\"1!R([uZ!fq$F0F)$\"1S`WJ\"H'H[F07%$\"1\"*************\\F0F )$\"1+++++++]F07%$\"1&f7bA&4%H'F0F)$\"1W`WJ\"H'H[F07%$\"1\"*********** *\\(F0F)$\"1*>A*=q7IVF07%$\"1IF$f!R`N&)F0F)$\"1YF$f!R`NNF07%$\"1(=A*=q 7I$*F0F)$\"16++++++DF07%$\"1R`WJ\"H'H)*F0F)$\"1=E^D_4%H\"F07%$\"\"\"F) F)$\"1Qj;_Da:;!#I7%$\"1X`WJ\"H'H)*F0F)$!1'e7bA&4%H\"F07%$\"1-A#*=q7I$* F0F)$!1$)***********\\#F07%$\"1_F$f!R`N&)F0F)$!1BF$f!R`NNF07%$\"1<++++ ++vF0F)$!1$=A*=q7IVF07%$\"1DE^D_4%H'F0F)$!1N`WJ\"H'H[F07%$\"1A++++++]F 0F)$!1+++++++]F07%$\"1?u[uZ!fq$F0F)$!1Z`WJ\"H'H[F07%$\"1A++++++DF0F)$! 11A#*=q7IVF07%$\"1#GnS4mWY\"F0F)$!1dF$f!R`NNF07%$\"18#y2\")H()p'F-F)$! 1D++++++DF07%$\"1mma&o3Pq\"F-F)$!1ME^D_4%H\"F07%$\"1]S9&4\"*R/\"!#WF)$ !1wEL/^3JKFbo7;F(7%$\"1XC%)>ilX;F-$\"1o.E^D_4W!#=F.7%$\"12IcFhZqkF-$\" 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