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}{CSTYLE "" -1 414 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 415 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Times" 0 14 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 0 11 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Unit 6: Matrix Algebra" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Chapter 3 1: Vectors as Arrows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 32 "Section 31.3: the cross product" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Copyright" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Co pyright * 2001 by Addison Wesley Longman, Inc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All rights reserved. No part of this publication may be reproduced, stored in a retrieval sys tem, or transmitted, in any form or by any means, electronic, mechanic al, photocopying, recording, or otherwise, without the prior written p ermission of the publisher. Printed in the United States of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "with(linalg):\nwith(plots) :\nwith(plottools):\nread(`pvac.txt`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 50 "Derivation \+ of Formula (12.5) for the Cross Product" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Formula (12.5) in Section 12. 4 can be derived from the three properties that " }{TEXT 256 1 "A" } {TEXT -1 1 " " }{TEXT 332 1 "x" }{TEXT -1 1 " " }{TEXT 257 1 "B" } {TEXT -1 35 ", the cross product of the vectors " }{TEXT 334 1 "A" } {TEXT -1 5 " and " }{TEXT 335 1 "B" }{TEXT -1 24 ", is orthogonal to b oth " }{TEXT 258 1 "A" }{TEXT -1 5 " and " }{TEXT 259 1 "B" }{TEXT -1 35 ", forms a right-handed system with " }{TEXT 336 1 "A" }{TEXT -1 5 " and " }{TEXT 337 1 "B" }{TEXT -1 25 ", and has length given by" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 3 "|| " } {TEXT 260 1 "A" }{TEXT -1 1 " " }{TEXT 333 1 "x" }{TEXT -1 1 " " } {TEXT 261 1 "B" }{TEXT -1 9 " || = || " }{TEXT 262 1 "A" }{TEXT -1 7 " || || " }{TEXT 263 1 "B" }{TEXT -1 8 " || sin " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 " \+ in " }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 22 " is the an gle between " }{TEXT 338 1 "A" }{TEXT -1 5 " and " }{TEXT 339 1 "B" } {TEXT -1 31 ". To do this, let the vectors " }{TEXT 340 1 "A" }{TEXT -1 2 ", " }{TEXT 341 1 "B" }{TEXT -1 6 ", and " }{TEXT 342 1 "C" } {TEXT -1 25 " be given respectively by" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "A := vector([seq(a[k], k=1..3)]):\nB := vector([seq(b[k],k=1..3)]):\nC := vector([seq(c[k],k= 1..3)]):\nprint(A,B,C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The condition that " }{TEXT 264 1 "C" }{TEXT -1 26 " be perpendicular to both " }{TEXT 265 1 "A" } {TEXT -1 5 " and " }{TEXT 266 1 "B" }{TEXT -1 25 " yields the two equa tions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "q1 := dotprod(A,C,orthogonal) = 0;\nq2 := dotprod(B,C ,orthogonal) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Then, " }}{PARA 259 "" 0 "" {TEXT -1 3 "|| " }{TEXT 343 1 "A" }{TEXT -1 1 " " }{TEXT 347 1 "x" }{TEXT -1 1 " " }{TEXT 344 1 "B" }{TEXT -1 9 " || = || " }{TEXT 345 1 "A" } {TEXT -1 7 " || || " }{TEXT 346 1 "B" }{TEXT -1 8 " || sin " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "yields the third equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " sqrt(c[1]^2+c[2]^2+c[3]^2) = sqrt(a[1]^2+a[2]^2+a[3]^2)*sqrt(b[1]^2+b[ 2]^2+b[3]^2)*sqrt(1-(a[1]*b[1]+a[2]*b[2]+a[3]*b[3])^2/(a[1]^2+a[2]^2+a [3]^2)/(b[1]^2+b[2]^2+b[3]^2))" "6#/-%%sqrtG6#,(*$&%\"cG6#\"\"\"\"\"#F ,*$&F*6#F-F-F,*$&F*6#\"\"$F-F,*(-F%6#,(*$&%\"aG6#F,F-F,*$&F;6#F-F-F,*$ &F;6#F4F-F,F,-F%6#,(*$&%\"bG6#F,F-F,*$&FH6#F-F-F,*$&FH6#F4F-F,F,-F%6#, &F,F,*(,(*&&F;6#F,F,&FH6#F,F,F,*&&F;6#F-F,&FH6#F-F,F,*&&F;6#F4F,&FH6#F 4F,F,F-,(*$&F;6#F,F-F,*$&F;6#F-F-F,*$&F;6#F4F-F,!\"\",(*$&FH6#F,F-F,*$ &FH6#F-F-F,*$&FH6#F4F-F,FhoFhoF," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "which we obtain in Maple \+ with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "normA := simplify(norm(A,2),symbolic);\nnormB := sim plify(norm(B,2),symbolic);\nnormC := simplify(norm(C,2),symbolic);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "q3 := normC = normA*normB*s in(angle(A,B));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Solving these three equations for \+ " }{XPPEDIT 18 0 "c[1],c[2],c[3]" "6%&%\"cG6#\"\"\"&F$6#\"\"#&F$6#\"\" $" }{TEXT -1 6 " gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "q4 := solve(\{q||(1..3)\},\{c[1],c[ 2],c[3]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "There are two solutions, namely," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C1 := subs(q4[1],op(C));\nC2 := subs(q4[2],op(C));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "MAT RIX([[a[2]*b[3]-a[3]*b[2]],[a[3]*b[1]-a[1]*b[3]],[a[1]*b[2]-a[2]*b[1]] ])" "6#-%'MATRIXG6#7%7#,&*&&%\"aG6#\"\"#\"\"\"&%\"bG6#\"\"$F.F.*&&F+6# F2F.&F06#F-F.!\"\"7#,&*&&F+6#F2F.&F06#F.F.F.*&&F+6#F.F.&F06#F2F.F87#,& *&&F+6#F.F.&F06#F-F.F.*&&F+6#F-F.&F06#F.F.F8" }{TEXT -1 12 " and \+ " }{XPPEDIT 18 0 "-MATRIX([[a[2]*b[3]-a[3]*b[2]],[a[3]*b[1]-a[1]*b[3 ]],[a[1]*b[2]-a[2]*b[1]]])" "6#,$-%'MATRIXG6#7%7#,&*&&%\"aG6#\"\"#\"\" \"&%\"bG6#\"\"$F/F/*&&F,6#F3F/&F16#F.F/!\"\"7#,&*&&F,6#F3F/&F16#F/F/F/ *&&F,6#F/F/&F16#F3F/F97#,&*&&F,6#F/F/&F16#F.F/F/*&&F,6#F.F/&F16#F/F/F9 F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "A simple test with the unit vectors \{" }{TEXT 351 1 "i" }{TEXT -1 2 ", " }{TEXT 352 1 "j" }{TEXT -1 2 ", " }{TEXT 353 1 "k" } {TEXT -1 267 "\} shows the first solution obeys the right-hand rule an d points in the appropriate direction. The other solution is its nega tive. Consequently, orthogonality, the length condition, and the righ t-hand rule determine the cross product which we write more compactly \+ as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 348 1 "A " }{TEXT -1 1 " " }{TEXT 350 1 "x" }{TEXT -1 1 " " }{TEXT 349 1 "B" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "det*MATRIX([[i, j, k], [a[1], a[2], a [3]], [b[1], b[2], b[3]]])" "6#*&%$detG\"\"\"-%'MATRIXG6#7%7%%\"iG%\"j G%\"kG7%&%\"aG6#F%&F06#\"\"#&F06#\"\"$7%&%\"bG6#F%&F:6#F4&F:6#F7F%" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "in terms of the determinant of a 3 " }{TEXT 354 1 "x" } {TEXT -1 64 " 3 matrix. (See Section 12.4 for a discussion of determi nants.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "In Maple, we could perform this test via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "subs(a[1]=1,a[2] =0,a[3]=0, b[1]=0,b[2]=1,b[3]=0, op(C1));\nsubs(a[1]=1,a[2]=0,a[3]=0, \+ b[1]=0,b[2]=1,b[3]=0, op(C2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "The first vector is co nsistent with the right-handed " }{TEXT 355 1 "i" }{TEXT -1 1 " " } {TEXT 358 1 "x" }{TEXT -1 1 " " }{TEXT 356 1 "j" }{TEXT -1 3 " = " } {TEXT 357 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "Alternatively, comparing to Maple's built -in " }{TEXT 267 9 "crossprod" }{TEXT -1 10 " command, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "crossp rod(A,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "is another way we might select the correc t solution from the two solutions found." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Example 31.6" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "If " }}{PARA 263 "" 0 "" {TEXT 359 1 "A" }{TEXT -1 5 " = 2 " }{TEXT 360 1 "i" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" } {TEXT -1 2 "2 " }{TEXT 361 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-`` " "6#,&%!G\"\"\"F$!\"\"" }{TEXT 362 1 "k" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 3 "and" }}{PARA 264 "" 0 "" {TEXT 363 1 "B" }{TEXT -1 3 " = " }{TEXT 364 1 "i" }{TEXT -1 3 " + " }{TEXT 365 1 "j" }{TEXT -1 3 " + " }{TEXT 366 1 "k" }{TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "that is, if" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A := vector([2, -2, -1]);\nB := vec tor([1, 1, 1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "then " }}{PARA 265 "" 0 "" {TEXT 367 1 "A" }{TEXT -1 1 " " }{TEXT 369 1 "x" }{TEXT -1 1 " " }{TEXT 368 1 "B" }{TEXT -1 7 " = det " }{XPPEDIT 18 0 "MATRIX([[i,j,k],[2,-1,-1], [1,1,1]])=MATRIX([[-1],[-3],[4]])" "6#/-%'MATRIXG6#7%7%%\"iG%\"jG%\"kG 7%\"\"#,$\"\"\"!\"\",$F/F07%F/F/F/-F%6#7%7#,$F/F07#,$\"\"$F07#\"\"%" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "AxB := c rossprod(A, B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "with " }{TEXT 370 1 "A" }{TEXT -1 2 ", " }{TEXT 371 1 "B" }{TEXT -1 6 ", and " }{TEXT 372 1 "A" }{TEXT -1 1 " " }{TEXT 374 1 "x" }{TEXT -1 1 " " }{TEXT 373 1 "B" }{TEXT -1 23 " shown in FIgure 31.10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 550 "p1 := arrow([0,0,0], A, [1, 1,1], .1,.3,.2, color=black):\np2 := arrow([0,0,0], B, [1,0,1], .1,.3, .2, color=black):\np3 := arrow([0,0,0], AxB, [2,1,1], .1,.3,.2, color= black):\np4 := textplot3d(\{[1,-1.5,-.7,`A`], [.5,.7,.9,`B`], [-.5,-1, 2.1,`AxB`]\}, color = black, font=[TIMES,ROMAN,14]):\np5 := spacecurve (\{[[0,0,0],[2,0,0],[2,-3,0],[0,-3,0],[0,0,0]], [[0,0,0],[0,0,4]]\}, c olor=black, linestyle=2):\ndisplay([p||(1..5)], orientation=[-30,75], \+ axes=boxed, scaling=constrained, labels=[`x `,` y`,`z `], labelf ont=[TIMES,ITALIC,12], tickmarks=[4,4,5]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 42 "Additional Properties of the Cross Product" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "A compendium of formulas for the cross product is listed in Table 31.3, the contents of which are the following." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "1. Scalar Multiplication \+ (" }{TEXT 311 1 "a" }{TEXT -1 1 " " }{TEXT 268 1 "A" }{TEXT -1 2 ") " }{TEXT 375 1 "x" }{TEXT -1 2 " (" }{TEXT 312 1 "b" }{TEXT -1 1 " " }{TEXT 269 1 "B" }{TEXT -1 4 ") = " }{TEXT 313 2 "ab" }{TEXT -1 2 " (" }{TEXT 270 1 "A" }{TEXT -1 1 " " }{TEXT 376 1 "x" }{TEXT -1 1 " \+ " }{TEXT 271 1 "B" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "2. Distributive Laws \+ " }{TEXT 272 1 "A" }{TEXT -1 1 " " }{TEXT 377 1 "x" }{TEXT -1 2 " (" }{TEXT 273 1 "B" }{TEXT -1 3 " + " }{TEXT 274 1 "C" }{TEXT -1 4 ") = \+ " }{TEXT 275 1 "A" }{TEXT -1 1 " " }{TEXT 378 1 "x" }{TEXT -1 1 " " } {TEXT 276 1 "B" }{TEXT -1 3 " + " }{TEXT 277 1 "A" }{TEXT -1 1 " " } {TEXT 379 1 "x" }{TEXT -1 1 " " }{TEXT 278 1 "C" }}{PARA 0 "" 0 "" {TEXT -1 53 " (" } {TEXT 279 1 "B" }{TEXT -1 3 " + " }{TEXT 280 1 "C" }{TEXT -1 2 ") " } {TEXT 380 1 "x" }{TEXT -1 1 " " }{TEXT 281 1 "A" }{TEXT -1 3 " = " } {TEXT 282 1 "B" }{TEXT -1 1 " " }{TEXT 381 1 "x" }{TEXT -1 1 " " } {TEXT 283 1 "A" }{TEXT -1 3 " + " }{TEXT 284 1 "C" }{TEXT -1 1 " " } {TEXT 382 1 "x" }{TEXT -1 1 " " }{TEXT 285 1 "A" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "3. Anti-Commutation \+ " }{TEXT 286 1 "B" }{TEXT -1 1 " " }{TEXT 383 1 "x" } {TEXT -1 1 " " }{TEXT 287 1 "A" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``-` `" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 288 1 "A" }{TEXT -1 1 " " }{TEXT 384 1 "x" }{TEXT -1 1 " " }{TEXT 289 1 "B" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "4. Non-Associative \+ " }{TEXT 290 1 "A" }{TEXT -1 1 " " }{TEXT 385 1 "x" }{TEXT -1 2 " (" }{TEXT 291 1 "B" }{TEXT -1 1 " " }{TEXT 386 1 "x" }{TEXT -1 1 " " }{TEXT 292 1 "C" }{TEXT -1 5 ") = (" }{TEXT 293 1 "A" }{TEXT 303 1 ". " }{TEXT 304 1 "C" }{TEXT -1 2 ") " }{TEXT 294 1 "B" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 2 " (" }{TEXT 295 1 "A" }{TEXT 305 1 "." }{TEXT 306 1 "B" }{TEXT -1 2 ") " }{TEXT 296 1 "C" }}{PARA 0 "" 0 "" {TEXT -1 54 " \+ (" }{TEXT 297 2 "A " }{TEXT 387 1 "x" }{TEXT 388 2 " B" }{TEXT -1 2 ") " }{TEXT 389 1 "x" }{TEXT -1 1 " " }{TEXT 298 1 "C" }{TEXT -1 4 " = (" }{TEXT 299 1 "A" }{TEXT 307 1 "." }{TEXT 308 1 "C" }{TEXT -1 2 ") " }{TEXT 300 1 "B" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 2 " (" }{TEXT 301 1 "B" }{TEXT 309 1 "." }{TEXT 310 1 "C" }{TEXT -1 2 ") " }{TEXT 302 1 "A" }} {PARA 0 "" 0 "" {TEXT -1 63 "_________________________________________ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 62 " \+ Table 31.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "The first two laws sugge st the cross product obeys some of the rules of ordinary algebra. How ever, the failure of the commutative law (3) and of the associative la ws (4) points out the singular nature of the cross product." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Additional Examples (Not in Te xt)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "The non-associativity of the cross product illustrated in the \+ following computation involving the vectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "A := vector([seq (a[k], k = 1..3)]):\nB := vector([seq(b[k], k = 1..3)]):\nC := vector( [seq(c[k], k = 1..3)]):\nprint(A,B,C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Computing " } {TEXT 314 1 "A" }{TEXT -1 4 " x (" }{TEXT 315 1 "B" }{TEXT -1 3 " x " }{TEXT 316 1 "C" }{TEXT -1 8 ") and (" }{TEXT 317 1 "A" }{TEXT -1 3 " x " }{TEXT 318 1 "B" }{TEXT -1 4 ") x " }{TEXT 319 1 "C" }{TEXT -1 9 ", we find" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 71 "q1 := crossprod(A, crossprod(B,C));\nq2 := crosspro d(crossprod(A,B), C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "That these are not equal is esta blished by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "map(simplify,evalm(q1-q2));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " } {TEXT 320 1 "A" }{TEXT -1 4 " x (" }{TEXT 321 1 "B" }{TEXT -1 3 " x " }{TEXT 322 1 "C" }{TEXT -1 5 ") = (" }{TEXT 323 1 "A" }{TEXT -1 3 " x \+ " }{TEXT 324 1 "B" }{TEXT -1 4 ") x " }{TEXT 325 1 "C" }{TEXT -1 68 ", then the difference just computed would have been the zero vector." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "On the o ther hand, compare " }{TEXT 326 1 "A" }{TEXT -1 4 " x (" }{TEXT 327 1 "B" }{TEXT -1 3 " x " }{TEXT 328 1 "C" }{TEXT -1 8 ") with " }{TEXT 390 1 "A" }{TEXT -1 1 " " }{TEXT 401 1 "x" }{TEXT -1 2 " (" }{TEXT 391 1 "B" }{TEXT -1 1 " " }{TEXT 402 1 "x" }{TEXT -1 1 " " }{TEXT 392 1 "C" }{TEXT -1 5 ") = (" }{TEXT 393 1 "A" }{TEXT 397 1 "." }{TEXT 398 1 "C" }{TEXT -1 2 ") " }{TEXT 394 1 "B" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 2 " (" }{TEXT 395 1 "A" }{TEXT 399 1 "." }{TEXT 400 1 "B" }{TEXT -1 2 ") " }{TEXT 396 1 "C" } {TEXT -1 15 ", that is, with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Q1 := evalm(dotprod(A,C,orth ogonal)*B - dotprod(A,B,orthogonal)*C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "via the subtr action" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(simplify,evalm(q1-Q1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "and comp are (" }{TEXT 329 1 "A" }{TEXT -1 3 " x " }{TEXT 330 1 "B" }{TEXT -1 4 ") x " }{TEXT 331 1 "C" }{TEXT -1 7 " with (" }{TEXT 403 2 "A " } {TEXT 413 1 "x" }{TEXT 414 2 " B" }{TEXT -1 2 ") " }{TEXT 415 1 "x" } {TEXT -1 1 " " }{TEXT 404 1 "C" }{TEXT -1 4 " = (" }{TEXT 405 1 "A" } {TEXT 409 1 "." }{TEXT 410 1 "C" }{TEXT -1 2 ") " }{TEXT 406 1 "B" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 2 " (" }{TEXT 407 1 "B" }{TEXT 411 1 "." }{TEXT 412 1 "C" }{TEXT -1 2 ") " }{TEXT 408 1 "A" }{TEXT -1 15 ", that is, with" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Q2 : = evalm(dotprod(A,C,orthogonal)*B - dotprod(B,C,orthogonal)*A);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "via the subtraction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(simplify,evalm(q2-Q2)); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }