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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "The cross product of two vectors " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "The definition of the cross product of two vectors " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 13 "cross product" }{TEXT -1 4 " or " }{TEXT 259 14 "vector product" }{TEXT -1 17 " of two vectors " }{TEXT 270 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 263 1 " i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 264 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "z[1]" "6#& %\"zG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 265 1 "k" }{TEXT -1 6 " and " } {TEXT 272 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " \+ = " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 1 " " }{TEXT 266 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "y[2];" "6#&%\"yG6#\"\"#" } {TEXT -1 1 " " }{TEXT 267 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "z[2] ;" "6#&%\"zG6#\"\"#" }{TEXT -1 1 " " }{TEXT 268 1 "k" }{TEXT -1 24 " i s the vector given by " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT 271 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 273 1 " " }{TEXT 299 1 "x" }{TEXT 298 2 " v" }{XPPEDIT 18 0 "``[2] = ` `(y[1]*z[2]-y[2]*z[1]);" "6#/&%!G6#\"\"#-F%6#,&*&&%\"yG6#\"\"\"F/&%\"z G6#F'F/F/*&&F-6#F'F/&F16#F/F/!\"\"" }{TEXT -1 1 " " }{TEXT 283 5 "i + " }{XPPEDIT 18 0 "``(x[2]*z[1]-x[1]*z[2]);" "6#-%!G6#,&*&&%\"xG6#\"\" #\"\"\"&%\"zG6#F,F,F,*&&F)6#F,F,&F.6#F+F,!\"\"" }{TEXT -1 1 " " } {TEXT 284 1 "j" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "``(x[1]*y[2]-x[2]*y [1]);" "6#-%!G6#,&*&&%\"xG6#\"\"\"F+&%\"yG6#\"\"#F+F+*&&F)6#F/F+&F-6#F +F+!\"\"" }{TEXT -1 1 " " }{TEXT 285 1 "k" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 37 "____________________________ _________" }{TEXT -1 4 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 198 "A convenient way construct this cross \+ product, especially when computing a cross product \"by hand\", is to \+ use the following symbolic determinant in which the first row consists of the basis vectors " }{TEXT 295 1 "i" }{TEXT -1 2 ", " }{TEXT 296 1 "j" }{TEXT -1 5 " and " }{TEXT 297 1 "k" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 4 "det " } {XPPEDIT 18 0 "matrix([[\"i\", \"j\", \"k\"], [x[1], y[1], z[1]], [x[2 ], y[2], z[2]]]);" "6#-%'matrixG6#7%7%Q\"i6\"Q\"jF)Q\"kF)7%&%\"xG6#\" \"\"&%\"yG6#F0&%\"zG6#F07%&F.6#\"\"#&F26#F:&F56#F:" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 " = det \+ " }{XPPEDIT 18 0 "matrix([[y[1],z[1]],[y[2],z[2]]])" "6#-%'matrixG6#7$ 7$&%\"yG6#\"\"\"&%\"zG6#F+7$&F)6#\"\"#&F-6#F2" }{TEXT -1 1 " " }{TEXT 289 1 "i" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 4 "d et " }{XPPEDIT 18 0 "matrix([[x[1],z[1]],[x[2] ,z[2]]])" "6#-%'matrixG 6#7$7$&%\"xG6#\"\"\"&%\"zG6#F+7$&F)6#\"\"#&F-6#F2" }{TEXT -1 1 " " } {TEXT 290 1 "j" }{TEXT -1 7 " + det " }{XPPEDIT 18 0 "matrix([[x[1],y[ 1]],[x[2] ,y[2]]])" "6#-%'matrixG6#7$7$&%\"xG6#\"\"\"&%\"yG6#F+7$&F)6# \"\"#&F-6#F2" }{TEXT -1 1 " " }{TEXT 291 1 "k" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = ``(y[1]*z[2]-y[2]*z[1]);" "6#/%!G-F$6#,&*&&%\"yG6# \"\"\"F,&%\"zG6#\"\"#F,F,*&&F*6#F0F,&F.6#F,F,!\"\"" }{TEXT -1 1 " " } {TEXT 286 1 "i" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-``(x[1]*z[2]-x[2]*z[ 1]);" "6#,$-%!G6#,&*&&%\"xG6#\"\"\"F,&%\"zG6#\"\"#F,F,*&&F*6#F0F,&F.6# F,F,!\"\"F6" }{TEXT -1 1 " " }{TEXT 287 1 "j" }{TEXT -1 4 " + " } {XPPEDIT 18 0 "``(x[1]*y[2]-x[2]*y[1]);" "6#-%!G6#,&*&&%\"xG6#\"\"\"F+ &%\"yG6#\"\"#F+F+*&&F)6#F/F+&F-6#F+F+!\"\"" }{TEXT -1 1 " " }{TEXT 288 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(y[1]*z[2]-y[2]*z[1]);" "6#/%!G-F$6#,&*&&%\"yG6#\"\"\"F,&%\"zG6#\"\"#F,F,*&&F*6#F0F,&F.6#F,F,! \"\"" }{TEXT -1 1 " " }{TEXT 292 5 "i + " }{XPPEDIT 18 0 "``(x[2]*z[1 ]-x[1]*z[2]);" "6#-%!G6#,&*&&%\"xG6#\"\"#\"\"\"&%\"zG6#F,F,F,*&&F)6#F, F,&F.6#F+F,!\"\"" }{TEXT -1 1 " " }{TEXT 293 1 "j" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "``(x[1]*y[2]-x[2]*y[1]);" "6#-%!G6#,&*&&%\"xG6#\"\"\"F +&%\"yG6#\"\"#F+F+*&&F)6#F/F+&F-6#F+F+!\"\"" }{TEXT -1 1 " " }{TEXT 294 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "wit h(linalg):\nx := 'x': y := 'y': z := 'z':\nM := matrix([[`\"i\"`,`\"j \"`,`\"k\"`],[x[1],y[1],z[1]],[x[2],y[2],z[2]]]);\n'det(M)'=sort(colle ct(det(M),[`\"i\"`,`\"j\"`,`\"k\"`]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7%%$\"i\"G%$\"j\"G%$\"k\"G7%&%\"xG6#\"\"\"&% \"yGF0&%\"zGF07%&F/6#\"\"#&F3F8&F5F8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$detG6#%\"MG,(*&,&*&&%\"yG6#\"\"\"F/&%\"zG6#\"\"#F/F/*&&F1F.F/&F- F2F/!\"\"F/%$\"i\"GF/F/*&,&*&&%\"xGF.F/F0F/F7*&F5F/&F=F2F/F/F/%$\"j\"G F/F/*&,&*&F \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 9 "Example I" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 280 1 "u" }{TEXT -1 5 " = 4 " }{TEXT 274 1 "i" }{TEXT -1 4 " -3 " }{TEXT 275 1 "j" } {TEXT -1 5 " + 2 " }{TEXT 276 1 "k" }{TEXT -1 5 " and " }{TEXT 281 1 " v" }{TEXT -1 5 " = 5 " }{TEXT 277 1 "i" }{TEXT -1 5 " + 2 " }{TEXT 278 1 "j" }{TEXT -1 3 " - " }{TEXT 279 1 "k" }{TEXT -1 6 " then " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 309 2 "u " }{TEXT 310 1 "x" } {TEXT 311 2 " v" }{TEXT -1 7 " = det " }{XPPEDIT 18 0 "matrix([[\"i\", \"j\", \"k\"], [4, -3, 2], [5, 2, -1]]);" "6#-%'matrixG6#7%7%Q\"i6\"Q \"jF)Q\"kF)7%\"\"%,$\"\"$!\"\"\"\"#7%\"\"&F1,$\"\"\"F0" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 " = \+ det " }{XPPEDIT 18 0 "matrix([[-3, 2], [2, -1]]);" "6#-%'matrixG6#7$7$ ,$\"\"$!\"\"\"\"#7$F+,$\"\"\"F*" }{TEXT -1 1 " " }{TEXT 303 1 "i" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 4 "det " }{XPPEDIT 18 0 "matrix([[4, 2], [5, -1]]);" "6#-%'matrixG6 #7$7$\"\"%\"\"#7$\"\"&,$\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 304 1 "j" }{TEXT -1 7 " + det " }{XPPEDIT 18 0 "matrix([[4, -3], [5, 2]]);" "6#- %'matrixG6#7$7$\"\"%,$\"\"$!\"\"7$\"\"&\"\"#" }{TEXT -1 1 " " }{TEXT 305 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(3-4);" "6#/%!G-F$6#,&\" \"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{TEXT 300 1 "i" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "-``(-4-10);" "6#,$-%!G6#,&\"\"%!\"\"\"#5F)F)" } {TEXT -1 1 " " }{TEXT 301 1 "j" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "``( 8+15);" "6#-%!G6#,&\"\")\"\"\"\"#:F(" }{TEXT -1 1 " " }{TEXT 302 1 "k " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -``;" "6#/%!G,$F$!\"\"" }{TEXT 306 3 "i " }{TEXT -1 5 "+ 14 " }{TEXT 307 1 "j" }{TEXT -1 7 " + 23 \+ " }{TEXT 308 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "M := matrix([[`\"i\"`,`\"j \"`,`\"k\"`],[4,-3,2],[5,2,-1]]);\n'det(M)'=sort(collect(det(M),[`\"i \"`,`\"j\"`,`\"k\"`]));\n``=[coeffs(rhs(%))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7%%$\"i\"G%$\"j\"G%$\"k\"G7%\"\"%! \"$\"\"#7%\"\"&F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$detG6#% \"MG,(%$\"i\"G!\"\"*&\"#9\"\"\"%$\"j\"GF-F-*&\"#BF-%$\"k\"GF-F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G7%!\"\"\"#9\"#B" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The cross product of tw o vectors can be calculated using the procedure " }{TEXT 0 9 "crosspro d" }{TEXT -1 8 " in the " }{TEXT 0 6 "linalg" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "x := 'x': y := 'y': z := 'z':\nv1 := vector([x[1],y[1],z[1]]); \nv2 := vector([x[2],y[2],z[2]]);\nlinalg[crossprod](v1,v2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G-%'vectorG6#7%&%\"xG6#\"\"\"&%\"yGF+&% \"zGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G-%'vectorG6#7%&%\"xG6# \"\"#&%\"yGF+&%\"zGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7 %,&*&&%\"yG6#\"\"\"F,&%\"zG6#\"\"#F,F,*&&F.F+F,&F*F/F,!\"\",&*&F2F,&% \"xGF/F,F,*&&F8F+F,F-F,F4,&*&F:F,F3F,F,*&F)F,F7F,F4" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "u := vecto r([4,-3,2]);\nv := vector([5,2,-1]);\nlinalg[crossprod](u,v); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'vectorG6#7%\"\"%!\"$\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7%\"\"&\"\"#!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%!\"\"\"#9\"#B" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 9 "crossprod" }{TEXT -1 69 " will accept lists as inpu t, although the output is still a vector.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "u := [4,-3,2];\nv \+ := [5,2,-1];\nlinalg[crossprod](u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"%!\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7% \"\"&\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%!\"\" \"#9\"#B" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Alternatively, the procedure " }{TEXT 0 12 "CrossProduct" }{TEXT -1 8 " in the " }{TEXT 0 13 "LinearAlgebra" }{TEXT -1 22 " package can be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 76 "u := < 4 | -3 | 2 >;\nv := < 5 | 2 | -1 >;\nLinearA lgebra[CrossProduct](u,v);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG -%'RTABLEG6$\"(Cy5&-%'VECTORG6#7%\"\"%!\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'RTABLEG6$\"(C " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 70 "The o rthogonality and anticommutative properties of the cross product " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "The cross product " }{TEXT 329 1 "v" }{XPPEDIT 18 0 "``[1]" "6# &%!G6#\"\"\"" }{TEXT 330 1 " " }{TEXT 332 1 "x" }{TEXT 331 2 " v" } {XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 17 " of two vectors \+ " }{TEXT 333 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 334 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" } {TEXT -1 4 " is " }{TEXT 259 18 "orthogonal to both" }{TEXT -1 1 " " } {TEXT 312 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " \+ and " }{TEXT 313 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "This can be checked directly \+ from the formula " }}{PARA 256 "" 0 "" {TEXT 314 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 315 1 " " }{TEXT 320 1 "x" }{TEXT 319 2 " v" }{XPPEDIT 18 0 "``[2] = ``(y[1]*z[2]-y[2]*z[1]);" "6#/&%!G6#\" \"#-F%6#,&*&&%\"yG6#\"\"\"F/&%\"zG6#F'F/F/*&&F-6#F'F/&F16#F/F/!\"\"" } {TEXT -1 1 " " }{TEXT 316 5 "i + " }{XPPEDIT 18 0 "``(x[2]*z[1]-x[1]* z[2]);" "6#-%!G6#,&*&&%\"xG6#\"\"#\"\"\"&%\"zG6#F,F,F,*&&F)6#F,F,&F.6# F+F,!\"\"" }{TEXT -1 1 " " }{TEXT 317 1 "j" }{TEXT -1 4 " + " } {XPPEDIT 18 0 "``(x[1]*y[2]-x[2]*y[1]);" "6#-%!G6#,&*&&%\"xG6#\"\"\"F+ &%\"yG6#\"\"#F+F+*&&F)6#F/F+&F-6#F+F+!\"\"" }{TEXT -1 1 " " }{TEXT 318 1 "k" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 41 "for the cros s product of the two vectors " }{TEXT 327 1 "v" }{XPPEDIT 18 0 "``[1] " "6#&%!G6#\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6# \"\"\"" }{TEXT -1 1 " " }{TEXT 321 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 322 1 "j" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 323 1 "k" }{TEXT -1 6 " and " }{TEXT 328 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x[2]; " "6#&%\"xG6#\"\"#" }{TEXT -1 1 " " }{TEXT 324 1 "i" }{TEXT -1 3 " + \+ " }{XPPEDIT 18 0 "y[2];" "6#&%\"yG6#\"\"#" }{TEXT -1 1 " " }{TEXT 325 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\"#" } {TEXT -1 1 " " }{TEXT 326 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " ( " }{TEXT 335 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 336 1 " " }{TEXT 338 1 " x" }{TEXT 337 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ) " }{TEXT 347 1 "." }{TEXT -1 1 " " }{TEXT 339 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 4 " = (" }{XPPEDIT 18 0 "``(y[1]*z[2]-y[2]*z[1]);" "6#-%!G6#,&*&&%\"yG6#\"\"\"F+&%\"zG6# \"\"#F+F+*&&F)6#F/F+&F-6#F+F+!\"\"" }{TEXT -1 1 " " }{TEXT 340 5 "i + " }{XPPEDIT 18 0 "``(x[2]*z[1]-x[1]*z[2]);" "6#-%!G6#,&*&&%\"xG6#\"\" #\"\"\"&%\"zG6#F,F,F,*&&F)6#F,F,&F.6#F+F,!\"\"" }{TEXT -1 1 " " } {TEXT 341 1 "j" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "``(x[1]*y[2]-x[2]*y [1]);" "6#-%!G6#,&*&&%\"xG6#\"\"\"F+&%\"yG6#\"\"#F+F+*&&F)6#F/F+&F-6#F +F+!\"\"" }{TEXT -1 1 " " }{TEXT 342 1 "k" }{TEXT -1 3 " ) " }{TEXT 346 1 "." }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" } {TEXT -1 1 " " }{TEXT 343 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "y[1] " "6#&%\"yG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 344 1 "j" }{TEXT -1 3 " + \+ " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 345 1 "k" }{TEXT -1 5 " ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``= ``(y[1]*z[2]-y[2]*z[1])*x[ 1]+``(x[2]*z[1]-x[1]*z[2])*y[1]+``(x[1]*y[2]-x[2]*y[1])*z[1]" "6#/%!G, (*&-F$6#,&*&&%\"yG6#\"\"\"F.&%\"zG6#\"\"#F.F.*&&F,6#F2F.&F06#F.F.!\"\" F.&%\"xG6#F.F.F.*&-F$6#,&*&&F:6#F2F.&F06#F.F.F.*&&F:6#F.F.&F06#F2F.F8F .&F,6#F.F.F.*&-F$6#,&*&&F:6#F.F.&F,6#F2F.F.*&&F:6#F2F.&F,6#F.F.F8F.&F0 6#F.F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x[1]*y[1]*z[2]-x[1]*y[2]*z[1 ]+x[2]*y[1]*z[1]-x[1]*y[1]*z[2]+x[1]*y[2]*z[1]-x[2]*y[1]*z[1]" "6#/%!G ,.*(&%\"xG6#\"\"\"F*&%\"yG6#F*F*&%\"zG6#\"\"#F*F**(&F(6#F*F*&F,6#F1F*& F/6#F*F*!\"\"*(&F(6#F1F*&F,6#F*F*&F/6#F*F*F**(&F(6#F*F*&F,6#F*F*&F/6#F 1F*F9*(&F(6#F*F*&F,6#F1F*&F/6#F*F*F**(&F(6#F1F*&F,6#F*F*&F/6#F*F*F9" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=0 " "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "The re lation ( " }{TEXT 348 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT 349 1 " " }{TEXT 351 1 "x" }{TEXT 350 2 " v" }{XPPEDIT 18 0 "``[ 2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ) " }{TEXT 353 1 "." }{TEXT -1 1 " \+ " }{TEXT 352 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 38 " = 0. can be checked in a similar way." }}{PARA 0 "" 0 "" {TEXT -1 103 "Alternatively, in terms of the symbolic determinant definition of the cross product it is evident that " }}{PARA 256 "" 0 "" {TEXT -1 4 " ( " }{TEXT 354 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT 355 1 " " }{TEXT 357 1 "x" }{TEXT 356 2 " v" }{XPPEDIT 18 0 "``[ 2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ) " }{TEXT 359 1 "." }{TEXT -1 1 " \+ " }{TEXT 358 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 7 " = det " }{XPPEDIT 18 0 "matrix([[x[1], y[1], z[1]], [x[1], y[1], z [1]], [x[2], y[2], z[2]]])" "6#-%'matrixG6#7%7%&%\"xG6#\"\"\"&%\"yG6#F +&%\"zG6#F+7%&F)6#F+&F-6#F+&F06#F+7%&F)6#\"\"#&F-6#F<&F06#F<" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }}{PARA 256 "" 0 "" {TEXT -1 4 " ( " }{TEXT 360 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\" \"\"" }{TEXT 361 1 " " }{TEXT 363 1 "x" }{TEXT 362 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ) " }{TEXT 365 1 "." } {TEXT -1 1 " " }{TEXT 364 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"# " }{TEXT -1 7 " = det " }{XPPEDIT 18 0 "matrix([[x[2], y[2], z[2]], [x [1], y[1], z[1]], [x[2], y[2], z[2]]])" "6#-%'matrixG6#7%7%&%\"xG6#\" \"#&%\"yG6#F+&%\"zG6#F+7%&F)6#\"\"\"&F-6#F5&F06#F57%&F)6#F+&F-6#F+&F06 #F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 102 "In each case the determinant has two equal rows, so, b y a standard property of determinants, we have " }}{PARA 256 "" 0 "" {TEXT -1 5 " ( " }{TEXT 366 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\" \"\"" }{TEXT 367 1 " " }{TEXT 369 1 "x" }{TEXT 368 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ) " }{TEXT 371 1 "." } {TEXT -1 1 " " }{TEXT 370 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\" " }{TEXT -1 15 " = 0 and ( " }{TEXT 372 1 "v" }{XPPEDIT 18 0 "``[1 ]" "6#&%!G6#\"\"\"" }{TEXT 373 1 " " }{TEXT 375 1 "x" }{TEXT 374 2 " v " }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ) " }{TEXT 377 1 "." }{TEXT -1 1 " " }{TEXT 376 1 "v" }{XPPEDIT 18 0 "``[2];" "6# &%!G6#\"\"#" }{TEXT -1 5 " = 0." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 400 26 "__________________________" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Another basic pro perty is that reversing the order of the two vectors changes the sign \+ of their cross product." }}{PARA 0 "" 0 "" {TEXT -1 12 "This is the " }{TEXT 259 16 "anti-commutative" }{TEXT -1 33 " property of the cross \+ product. " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 378 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 379 1 " " }{TEXT 381 1 " x" }{TEXT 380 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" } {XPPEDIT 18 0 "``= -``" "6#/%!G,$F$!\"\"" }{TEXT 382 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT 383 1 " " }{TEXT 385 1 "x" } {TEXT 384 2 " v" }{XPPEDIT 18 0 "``[1];" "6#&%!G6#\"\"\"" }{TEXT -1 4 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 401 12 "____________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "To see this compare" }}{PARA 256 "" 0 "" {TEXT 386 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\" \"\"" }{TEXT 387 1 " " }{TEXT 392 1 "x" }{TEXT 391 2 " v" }{XPPEDIT 18 0 "``[2] = ``(y[1]*z[2]-y[2]*z[1]);" "6#/&%!G6#\"\"#-F%6#,&*&&%\"yG 6#\"\"\"F/&%\"zG6#F'F/F/*&&F-6#F'F/&F16#F/F/!\"\"" }{TEXT -1 1 " " } {TEXT 388 5 "i + " }{XPPEDIT 18 0 "``(x[2]*z[1]-x[1]*z[2]);" "6#-%!G6 #,&*&&%\"xG6#\"\"#\"\"\"&%\"zG6#F,F,F,*&&F)6#F,F,&F.6#F+F,!\"\"" } {TEXT -1 1 " " }{TEXT 389 1 "j" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "``( x[1]*y[2]-x[2]*y[1]);" "6#-%!G6#,&*&&%\"xG6#\"\"\"F+&%\"yG6#\"\"#F+F+* &&F)6#F/F+&F-6#F+F+!\"\"" }{TEXT -1 1 " " }{TEXT 390 1 "k" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "with " }}{PARA 256 "" 0 "" {TEXT 393 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT 394 1 " " }{TEXT 399 1 "x" }{TEXT 398 2 " v" }{XPPEDIT 18 0 "``[1] = ``(y[2]* z[1]-y[1]*z[2]);" "6#/&%!G6#\"\"\"-F%6#,&*&&%\"yG6#\"\"#F'&%\"zG6#F'F' F'*&&F-6#F'F'&F16#F/F'!\"\"" }{TEXT -1 1 " " }{TEXT 395 5 "i + " } {XPPEDIT 18 0 "``(x[1]*z[2]-x[2]*z[2]);" "6#-%!G6#,&*&&%\"xG6#\"\"\"F+ &%\"zG6#\"\"#F+F+*&&F)6#F/F+&F-6#F/F+!\"\"" }{TEXT -1 1 " " }{TEXT 396 1 "j" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "``(x[2]*y[1]-x[1]*y[2]); " "6#-%!G6#,&*&&%\"xG6#\"\"#\"\"\"&%\"yG6#F,F,F,*&&F)6#F,F,&F.6#F+F,! \"\"" }{TEXT -1 1 " " }{TEXT 397 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Geometrical interpretation of the cross product " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 18 "The cross product " }{TEXT 402 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 403 1 " " }{TEXT 405 1 " x" }{TEXT 404 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 40 " of two (none-parallel) vectors vectors " }{TEXT 406 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 407 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 4 " is " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 409 1 "v" }{XPPEDIT 18 0 "``[ 1]" "6#&%!G6#\"\"\"" }{TEXT 410 1 " " }{TEXT 412 1 "x" }{TEXT 411 2 " \+ v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 10 " = ( || " } {TEXT 408 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 8 " \+ || || " }{TEXT 413 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" } {TEXT -1 4 " || " }{XPPEDIT 18 0 "sin(theta)" "6#-%$sinG6#%&thetaG" } {TEXT -1 4 " ) " }{TEXT 414 1 "n" }{TEXT -1 2 ", " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 420 21 "_____________________" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 22 " is the angle bet ween " }{TEXT 416 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 5 " and " }{TEXT 417 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\" \"#" }{TEXT -1 6 ", and " }{TEXT 415 1 "n" }{TEXT -1 46 " is a unit ve ctor which is orthogonal to both " }{TEXT 418 1 "v" }{XPPEDIT 18 0 "`` [1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 419 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{TEXT 457 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 5 " and " }{TEXT 458 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\" \"#" }{TEXT -1 20 " are parallel, then " }{TEXT 453 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 454 1 " " }{TEXT 456 1 "x" } {TEXT 455 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 22 " is the zero vector. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 500 267 267 {PLOTDATA 2 "65-%'CURVESG6&7$7$$\"\"!F)F(7$$\"\" $F)$\"\"\"F)7%7$$\"+7')=%*G!\"*$\"+lTV<5F3F*7$$\"+)Q6e#HF3$\"+^$ecA*!# 5-%&STYLEG6#%,PATCHNOGRIDG-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6&7$F '7$$\"\"&F)F(7%7$$\"+++++\\F3$\"+++++]!#6FJ7$FO$!+++++]FSF<-%&COLORG6& FCF)$\"\"(!\"\"F)-F$6'7$F'7$F($\"#AFfn7%7$FU$\"++++M@F3Fjn7$FQF_oF<-FA 6&FCF(F(FD-%*THICKNESSG6#\"\"#-%)POLYGONSG6%7%F'F*FJ-FX6&FC$\"#'*!\"#F ^pF^pF<-F$6%7$F*FJ-FA6&FC$\")=THvFFFfpFfp-%*LINESTYLEGFfo-F$6$7+7$F-F( 7$$\"3a+++\")p%>'**!#=$\"3G+++6(Hi[$!#>7$$\"33+++Iv2[)*Faq$\"3e*****z5 Ff%pFdq7$$\"37+++j#e#f'*Faq$\"3'******4=w_.\"Faq7$$\"3W+++3i#pR*Faq$\" 3))*****Rd!3o8Faq7$$\"3K+++qy2j!*Faq$\"37+++ZIZ!p\"Faq7$$\"3I+++PSDg') Faq$\"35+++++++?Faq7$$\"3%)*****>W?:>)Faq$\"3#******fu0VH#Faq7$$\"3k** ****HWWgwFaq$\"31+++R/:rDFaq-FA6&FCF)F)F)-F$6$7>7$$\"3;+++-+++]Faq$\"3 !******z$)*e.5!#<7$$\"3%)*****>Okdt&Faq$\"32+++#=RB-\"Fbt7$$\"3C+++&4w yU'Faq$\"3.+++BAeV5Fbt7$$\"3=+++5y1rqFaq$\"3)******zGdr1\"Fbt7$Fds$\"3 -+++c\\)G4\"Fbt7$F_s$\"3,+++D%p07\"Fbt7$Fjr$\"3#*************\\6Fbt7$F er$\"3++++&p_4=\"Fbt7$F`r$\"3*******H%>>87Fbt7$F[r$\"35+++#QskC\"Fbt7$ Ffq$\"3%*******G2a!G\"Fbt7$F_q$\"3*******HqP^J\"Fbt7$F-$\"33++++++]8Fb t7$F_q$\"3'******pHi[Q\"Fbt7$Ffq$\"3-+++r#f%>9Fbt7$F[r$\"33+++=w_`9Fbt 7$F`r$\"3'******p03o[\"Fbt7$Fer$\"3'******\\IZ!>:Fbt7$Fjr$\"3/++++++]: Fbt7$F_s$\"3%******\\dI%z:Fbt7$Fds$\"3%******R/:rg\"Fbt7$F^u$\"3)***** *>rUGj\"Fbt7$Fit$\"3#******px)e#Faq$\"3'*******H.PO " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 75 "Properties of the c ross product following from the geometrical description " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 44 "W e consider the consequences of the formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 460 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\" " }{TEXT 461 1 " " }{TEXT 463 1 "x" }{TEXT 462 2 " v" }{XPPEDIT 18 0 " ``[2];" "6#&%!G6#\"\"#" }{TEXT -1 10 " = ( || " }{TEXT 459 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 8 " || || " }{TEXT 464 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 4 " || " } {XPPEDIT 18 0 "sin(theta)" "6#-%$sinG6#%&thetaG" }{TEXT -1 4 " ) " } {TEXT 465 1 "n" }{TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 37 "for the cross product of two vectors " }{TEXT 471 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 472 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 22 " is the angle between " }{TEXT 467 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 468 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" } {TEXT -1 6 ", and " }{TEXT 466 1 "n" }{TEXT -1 46 " is a unit vector w hich is orthogonal to both " }{TEXT 469 1 "v" }{XPPEDIT 18 0 "``[1]" " 6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 470 1 "v" }{XPPEDIT 18 0 "` `[2]" "6#&%!G6#\"\"#" }{TEXT -1 106 ". in an effort to show that this \+ definition is equivalent to the definition given in terms of coordinat es." }}{PARA 0 "" 0 "" {TEXT -1 57 "It is easy to check from either of the definitions that: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 564 1 "i" }{TEXT -1 1 " " }{TEXT 562 1 "x" }{TEXT -1 1 " " }{TEXT 563 1 "i" }{TEXT -1 5 " = " }{TEXT 565 1 "j" }{TEXT -1 1 " " }{TEXT 566 1 "x" }{TEXT -1 1 " " }{TEXT 567 1 "j" }{TEXT -1 5 " = " }{TEXT 568 1 "k" }{TEXT -1 1 " " }{TEXT 569 1 "x" }{TEXT -1 1 " " }{TEXT 570 1 "k " }{TEXT -1 4 " = " }{TEXT 571 1 "0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 476 1 "i" } {TEXT -1 1 " " }{TEXT 474 1 "x" }{TEXT -1 1 " " }{TEXT 475 1 "j" } {TEXT -1 5 " = " }{TEXT 491 1 "k" }{XPPEDIT 18 0 "``= -``" "6#/%!G,$ F$!\"\"" }{TEXT -1 1 " " }{TEXT 477 1 "j" }{TEXT -1 1 " " }{TEXT 473 1 "x" }{TEXT -1 1 " " }{TEXT 478 1 "i" }{TEXT -1 2 ", " }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{TEXT 485 1 "j" }{TEXT -1 1 " " }{TEXT 480 1 " x" }{TEXT -1 1 " " }{TEXT 481 1 "k" }{TEXT -1 5 " = " }{TEXT 492 1 " i" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= -``" "6#/%!G,$F$!\"\"" }{TEXT -1 1 " " }{TEXT 486 1 "k" }{TEXT -1 1 " " }{TEXT 479 1 "x" }{TEXT -1 1 " " }{TEXT 487 1 "j" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 493 1 "k" }{TEXT -1 1 " " }{TEXT 483 1 "x" }{TEXT -1 1 " " }{TEXT 494 1 "i" }{TEXT -1 5 " = " }{TEXT 495 1 "j" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``= -``" "6#/%!G,$F$!\"\"" }{TEXT -1 1 " " }{TEXT 496 1 "i" }{TEXT -1 1 " " }{TEXT 482 1 "x" }{TEXT -1 1 " " }{TEXT 484 1 "k" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 149 130 130 {PLOTDATA 2 "6.-%'CURVESG6$777$$\"3W+++3i#pR*!#=$!3w*****R V,-U$F*7$$\"37+++j#e#f'*F*$!3')*****>X!>)e#F*7$$\"33+++Iv2[)*F*$!3*)** ***px\"[O'**F*$!3E+++xUd:()!#>7$$\"\"\"\"\"!$F@F@7$ F8$\"3E+++xUd:()F<7$F3$\"3*)*****px\"[OX!>)e#F*7$F( $\"3w*****RV,-U$F*7$$\"3K+++qy2j!*F*$\"31+++=E=EUF*7$$\"3I+++PSDg')F*$ \"33+++,+++]F*7$$\"3%)*****>W?:>)F*$\"3++++kVwNdF*7$$\"3k******HWWgwF* $\"3[+++)4wyU'F*7$$\"3=+++5y1rqF*$\"3U+++8y1rqF*7$$\"3C+++&4wyU'F*$\"3 !)*****>VW/m(F*7$$\"3%)*****>Okdt&F*$\"3-+++W/_\">)F*7$$\"3;+++-+++]F* FT7$$\"3*)*****fh#=EUF*$\"3S+++ry2j!*F*7$$\"3#)*****zU,-U$F*$\"3]***** *4i#pR*F*7$$\"3y*****4X!>)e#F*F.7$$\"37+++t<[O)e#F*$\"3/+++i #e#f'*F*7$$!3z*****4W,-U$F*$\"3=+++0i#pR*F*7$$!39+++>E=EUF*$\"3C+++py2 j!*F*7$$!3T+++0+++]F*$\"39+++NSDg')F*7$$!3b*****>Pkdt&F*$\"3W*****pV?: >)F*7$$!3a*******4wyU'F*$\"3c******GWWgwF*7$$!3s*****z\"y1rqF*$\"3%)** ***f!y1rqF*7$$!3))*****HVW/m(F*Fbo7$$!3=+++Y/_\">)F*$\"3f******eVwNdF* 7$$!3o*****H/a-m)F*$\"3!)*****4*******\\F*7$$!3[+++sy2j!*F*$\"3t*****R h#=EUF*7$$!3e*****4@EpR*F*$\"3@+++E9??MF*7$$!37+++j#e#f'*F*$\"3;+++\\/ >)e#F*7$$!3;+++Jv2[)*F*$\"3&******4x\"[O)p%>'**F*$\"31+ ++uTd:()F<7$$!\"\"F@$!3!)*****\\hn?5%!#F7$$!3Y+++!)p%>'**F*$!3')*****f Nubr)F<7$$!3#******z_x![)*F*$!30+++*y\"[O)e#F*7$$!35+++/i#pR*F*$!3&******HW,-U$F*F`q-F$6$777$$!3]+++FWWgwF *$!3q*****>5wyU'F*7$$!3w*****\\!y1rqF*$!3#)******=y1rqF*7$$!3\\*****f3 wyU'F*$!3Y+++SWWgwF*7$$!3`+++dVwNdF*$!3M+++[/_\">)F*7$$!3?+++*)****** \\F*$!3w*****R/a-m)F*7$$!37+++7E=EUF*$!3Y*****H(y2j!*F*7$$!3/+++C9??MF *Fau7$$!3u*****pV!>)e#F*$!3Y+++n#e#f'*F*7$$!31+++p<[OU,J:'FivFev7$$\"38+++wVd:()F)e#F*Few7$$\"37+++X9??MF*Fjw7 $$\"36+++KE=EUF*$!3#)*****R'y2j!*F*7$$\"3j******3+++]F*$!3'******H.a-m )F*7$$\"3!)*****\\Pkdt&F*$!3Q+++N/_\">)F*7$$\"3z*****H5wyU'F*$!3U+++EW WgwF*7$$\"3!*******>y1rqF*$!3g*****H!y1rqF*7$$\"3_*****>WW/m(F*$!3W+++ %3wyU'F*F`q-F$6$7%7$$\"3,++++++55!#<$!3))**************HF*7$$\"3y***** ******pR*F*$!3E++++++?MF*7$$\"3A+++++++!*F*$!3++++++++DF*F`q-F$6$7%7$$ !3;+++++++xF*$!3#)*************H(F*7$$!39++++++gwF*$!3;++++++IkF*7$$!3 Y**************oF*$!3%**************>'F*F`q-F$6$7%7$$!35+++++++BF*$\"3 I+++++++\"*F*7$$!3))************RQ\"i6\"-Faq6&FcqF@F@F@-%%FONTG6%%*HELV ETICAG%%BOLDG\"#7-Fbal6&7$$\"+SSDg')!#5$!\"&FfvQ\"jFfalFgalFial-Fbal6& 7$$!+SSDg')FdblFeblQ\"kFfalFgalFial-%+AXESLABELSG6%Q!FfalFacl-Fjal6#%( DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$FdclFdcl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}{PARA 0 " " 0 "" {TEXT -1 51 "Taking the cross product of any two of the vectors " }{TEXT 488 1 "i" }{TEXT -1 2 ", " }{TEXT 489 1 "j" }{TEXT -1 5 " an d " }{TEXT 490 1 "k" }{TEXT -1 101 " in the cyclical order suggested b y the diagram gives the next vector in this order with a plus sign." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "The cros s product defined from equation (i) has the following properties." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 259 20 "anti-commutative law" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 528 1 "u" }{TEXT -1 1 " " }{TEXT 527 1 "x" } {TEXT -1 1 " " }{TEXT 529 1 "v" }{XPPEDIT 18 0 "`` = -``" "6#/%!G,$F$! \"\"" }{TEXT 532 1 "v" }{TEXT -1 1 " " }{TEXT 530 1 "x" }{TEXT -1 1 " \+ " }{TEXT 531 1 "u" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 15 "assoc iative law" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 4 " (r " } {TEXT 497 1 "u" }{TEXT -1 1 ")" }{TEXT 500 1 " " }{TEXT 499 1 "x" } {TEXT 498 1 " " }{TEXT -1 3 "(s " }{TEXT 501 1 "v" }{TEXT -1 1 ")" } {TEXT 503 1 " " }{TEXT -1 10 " = (rs) ( " }{TEXT 504 1 "u" }{TEXT -1 1 " " }{TEXT 502 1 "x" }{TEXT -1 1 " " }{TEXT 505 1 "v" }{TEXT -1 18 " ) ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 36 "follows easily from the formula (i)." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 16 "distributive law" }{TEXT -1 1 ":" }} {PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 507 1 "u" }{TEXT -1 1 " " } {TEXT 506 1 "x" }{TEXT -1 3 " ( " }{TEXT 508 1 "v" }{TEXT -1 3 " + " } {TEXT 515 1 "w" }{TEXT -1 7 " ) = ( " }{TEXT 510 1 "u" }{TEXT -1 1 " \+ " }{TEXT 509 1 "x" }{TEXT -1 1 " " }{TEXT 511 1 "v" }{TEXT -1 7 " ) + \+ ( " }{TEXT 513 1 "u" }{TEXT -1 1 " " }{TEXT 512 1 "x" }{TEXT -1 1 " " }{TEXT 514 1 "w" }{TEXT -1 16 " ) ------- (iv) " }}{PARA 0 "" 0 "" {TEXT -1 59 "is harder to demonstrate, but we shall consider this late r." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "On ce (iv) is established, the companion " }{TEXT 259 16 "distributive la w" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 7 " ( " }{TEXT 516 1 "v" }{TEXT -1 3 " + " }{TEXT 522 1 "w" }{TEXT -1 3 " ) " }{TEXT 523 1 "x" }{TEXT -1 1 " " }{TEXT 524 1 "u" }{TEXT -1 5 " = ( " }{TEXT 518 1 "v" }{TEXT -1 1 " " }{TEXT 517 1 "x" }{TEXT -1 1 " " }{TEXT 519 1 "u" }{TEXT -1 7 " ) + ( " }{TEXT 521 2 "w " }{TEXT 520 1 "x" }{TEXT 525 1 " " }{TEXT 526 1 "u" }{TEXT -1 16 " ) ------- (v) " }}{PARA 0 " " 0 "" {TEXT -1 44 "follows by multiplying both sides of (iv) by" } {XPPEDIT 18 0 "``-1" "6#,&%!G\"\"\"F%!\"\"" }{TEXT -1 22 ", and then u sing (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "The properties (ii) to (v) above, together with the rules for \+ forming cross products of the unit vectors " }{TEXT 545 1 "i" }{TEXT -1 2 ", " }{TEXT 546 1 "j" }{TEXT -1 5 " and " }{TEXT 547 1 "k" } {TEXT -1 90 ", are sufficient for the computation of cross products of vectors given in terms of their " }{TEXT 542 1 "i" }{TEXT -1 2 ", " } {TEXT 543 1 "j" }{TEXT -1 2 ", " }{TEXT 544 1 "k" }{TEXT -1 13 " compo nents. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 533 7 "Example" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {TEXT 540 1 "u" }{TEXT -1 5 " = 4 " }{TEXT 534 1 "i" }{TEXT -1 4 " -3 \+ " }{TEXT 535 1 "j" }{TEXT -1 5 " + 2 " }{TEXT 536 1 "k" }{TEXT -1 5 " \+ and " }{TEXT 541 1 "v" }{TEXT -1 5 " = 5 " }{TEXT 537 1 "i" }{TEXT -1 5 " + 2 " }{TEXT 538 1 "j" }{TEXT -1 3 " - " }{TEXT 539 1 "k" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{TEXT 549 1 "u" } {TEXT -1 1 " " }{TEXT 548 1 "x" }{TEXT -1 1 " " }{TEXT 550 1 "v" } {TEXT -1 7 " = ( 4 " }{TEXT 551 1 "i" }{TEXT -1 4 " -3 " }{TEXT 552 1 "j" }{TEXT -1 5 " + 2 " }{TEXT 553 1 "k" }{TEXT -1 3 " ) " }{TEXT 557 1 "x" }{TEXT -1 5 " ( 5 " }{TEXT 554 1 "i" }{TEXT -1 5 " + 2 " }{TEXT 555 1 "j" }{TEXT -1 3 " - " }{TEXT 556 1 "k" }{TEXT -1 4 " ). " }} {PARA 0 "" 0 "" {TEXT -1 60 "Using the associative and distributive pr operties we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 559 1 " u" }{TEXT -1 1 " " }{TEXT 558 1 "x" }{TEXT -1 1 " " }{TEXT 560 1 "v" } {TEXT -1 8 " = 20 ( " }{TEXT 561 1 "i" }{TEXT -1 1 " " }{TEXT 572 1 "x " }{TEXT -1 1 " " }{TEXT 573 1 "i" }{TEXT -1 9 " ) + 8 ( " }{TEXT 574 1 "i" }{TEXT -1 1 " " }{TEXT 575 1 "x" }{TEXT -1 1 " " }{TEXT 576 1 "j " }{TEXT -1 2 " )" }{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\"F$!\"\"" } {TEXT -1 4 "4 ( " }{TEXT 577 1 "i" }{TEXT -1 1 " " }{TEXT 578 1 "x" } {TEXT -1 1 " " }{TEXT 588 1 "k" }{TEXT -1 3 " ) " }{XPPEDIT 18 0 "-15 " "6#,$\"#:!\"\"" }{TEXT -1 3 " ( " }{TEXT 586 1 "j" }{TEXT -1 1 " " } {TEXT 579 1 "x" }{TEXT -1 1 " " }{TEXT 580 1 "i" }{TEXT -1 2 " )" } {XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 3 "6( " } {TEXT 587 1 "j" }{TEXT -1 1 " " }{TEXT 581 1 "x" }{TEXT -1 1 " " } {TEXT 582 1 "j" }{TEXT -1 9 " ) + 3 ( " }{TEXT 584 1 "j" }{TEXT -1 1 " " }{TEXT 583 1 "x" }{TEXT -1 1 " " }{TEXT 585 1 "k" }{TEXT -1 10 " ) \+ + 10 ( " }{TEXT 634 1 "k" }{TEXT -1 1 " " }{TEXT 589 1 "x" }{TEXT -1 1 " " }{TEXT 590 1 "i" }{TEXT -1 9 " ) + 4 ( " }{TEXT 591 1 "k" } {TEXT -1 1 " " }{TEXT 592 1 "x" }{TEXT -1 1 " " }{TEXT 593 1 "j" } {TEXT -1 3 " ) " }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" }{TEXT -1 3 " ( \+ " }{TEXT 594 1 "k" }{TEXT -1 1 " " }{TEXT 595 1 "x" }{TEXT -1 1 " " } {TEXT 596 1 "k" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "Using the relations: " }{TEXT 620 1 "i" } {TEXT -1 1 " " }{TEXT 618 1 "x" }{TEXT -1 1 " " }{TEXT 619 1 "i" } {TEXT -1 5 " = " }{TEXT 621 1 "j" }{TEXT -1 1 " " }{TEXT 622 1 "x" } {TEXT -1 1 " " }{TEXT 623 1 "j" }{TEXT -1 5 " = " }{TEXT 624 1 "k" } {TEXT -1 1 " " }{TEXT 625 1 "x" }{TEXT -1 1 " " }{TEXT 626 1 "k" } {TEXT -1 4 " = " }{TEXT 627 1 "0" }{TEXT -1 3 ", " }{TEXT 600 1 "i" }{TEXT -1 1 " " }{TEXT 598 1 "x" }{TEXT -1 1 " " }{TEXT 599 1 "j" } {TEXT -1 5 " = " }{TEXT 612 1 "k" }{XPPEDIT 18 0 "``= -``" "6#/%!G,$ F$!\"\"" }{TEXT -1 1 " " }{TEXT 601 1 "j" }{TEXT -1 1 " " }{TEXT 597 1 "x" }{TEXT -1 1 " " }{TEXT 602 1 "i" }{TEXT -1 3 ", " }{TEXT 609 1 "j" }{TEXT -1 1 " " }{TEXT 604 1 "x" }{TEXT -1 1 " " }{TEXT 605 1 "k" }{TEXT -1 5 " = " }{TEXT 613 1 "i" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `= -``" "6#/%!G,$F$!\"\"" }{TEXT -1 1 " " }{TEXT 610 1 "k" }{TEXT -1 1 " " }{TEXT 603 1 "x" }{TEXT -1 1 " " }{TEXT 611 1 "j" }{TEXT -1 3 ", " }{TEXT 614 1 "k" }{TEXT -1 1 " " }{TEXT 607 1 "x" }{TEXT -1 1 " " }{TEXT 615 1 "i" }{TEXT -1 5 " = " }{TEXT 616 1 "j" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "``= -``" "6#/%!G,$F$!\"\"" }{TEXT -1 1 " " }{TEXT 617 1 "i" }{TEXT -1 1 " " }{TEXT 606 1 "x" }{TEXT -1 1 " " }{TEXT 608 1 "k" }{TEXT -1 7 " gives:" }}{PARA 256 "" 0 "" {TEXT 629 1 "u" } {TEXT -1 1 " " }{TEXT 628 1 "x" }{TEXT -1 1 " " }{TEXT 630 1 "v" } {TEXT -1 5 " = 8 " }{TEXT 633 1 "k" }{TEXT -1 1 " " }{XPPEDIT 18 0 "- \+ 4" "6#,$\"\"%!\"\"" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "``" "6#%!G" } {XPPEDIT 18 0 "-``" "6#,$%!G!\"\"" }{TEXT 636 1 "j" }{TEXT -1 3 " ) " }{XPPEDIT 18 0 "-15" "6#,$\"#:!\"\"" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "` ` " "6#%!G" }{XPPEDIT 18 0 "-``" "6#,$%!G!\"\"" }{TEXT 729 1 "k" } {TEXT -1 9 " ) + 3 ( " }{TEXT 631 1 "i" }{TEXT -1 10 " ) + 10 ( " } {TEXT 632 1 "j" }{TEXT -1 7 " ) + 4(" }{XPPEDIT 18 0 "``-``" "6#,&%!G \"\"\"F$!\"\"" }{TEXT 635 1 "i" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "``- ``" "6#,&%!G\"\"\"F$!\"\"" } {TEXT 637 1 "i" }{TEXT -1 6 " + 14 " }{TEXT 638 1 "j" }{TEXT -1 6 " + \+ 23 " }{TEXT 639 1 "k" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 109 "We can use the method of this exam ple to derive the formula for the cross product given in the first sec tion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "L et " }{TEXT 646 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 640 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG 6#\"\"\"" }{TEXT -1 1 " " }{TEXT 641 1 "j" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 642 1 " k" }{TEXT -1 6 " and " }{TEXT 647 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&% !G6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 1 " " }{TEXT 643 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "y[2 ];" "6#&%\"yG6#\"\"#" }{TEXT -1 1 " " }{TEXT 644 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\"#" }{TEXT -1 1 " " }{TEXT 645 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " } {TEXT 648 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 649 1 " " }{TEXT 651 1 "x" }{TEXT 650 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6 #\"\"#" }{TEXT -1 5 " = ( " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" } {TEXT -1 1 " " }{TEXT 652 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "y[1] " "6#&%\"yG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 653 1 "j" }{TEXT -1 3 " + \+ " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 654 1 "k" }{TEXT -1 3 " ) " }{TEXT 658 1 "x" }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 1 " " }{TEXT 655 1 "i" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "y[2];" "6#&%\"yG6#\"\"#" }{TEXT -1 1 " " }{TEXT 656 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "z[2];" "6#&%\"z G6#\"\"#" }{TEXT -1 1 " " }{TEXT 657 1 "k" }{TEXT -1 3 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Using the assoc iative and distributive properties we have: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 659 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\" \"" }{TEXT 660 1 " " }{TEXT 662 1 "x" }{TEXT 661 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x[1]*x[2] " "6#*&&%\"xG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT -1 3 " ( " }{TEXT 663 1 "i " }{TEXT -1 1 " " }{TEXT 664 1 "x" }{TEXT -1 1 " " }{TEXT 665 1 "i" } {TEXT -1 5 " ) + " }{XPPEDIT 18 0 "x[1]*y[2]" "6#*&&%\"xG6#\"\"\"F'&% \"yG6#\"\"#F'" }{TEXT -1 3 " ( " }{TEXT 666 1 "i" }{TEXT -1 1 " " } {TEXT 667 1 "x" }{TEXT -1 1 " " }{TEXT 668 1 "j" }{TEXT -1 5 " ) + " } {XPPEDIT 18 0 "x[1]*z[2]" "6#*&&%\"xG6#\"\"\"F'&%\"zG6#\"\"#F'" } {TEXT -1 3 " ( " }{TEXT 669 1 "i" }{TEXT -1 1 " " }{TEXT 670 1 "x" } {TEXT -1 1 " " }{TEXT 680 1 "k" }{TEXT -1 5 " ) + " }{XPPEDIT 18 0 "y[ 1]*x[2]" "6#*&&%\"yG6#\"\"\"F'&%\"xG6#\"\"#F'" }{TEXT -1 3 " ( " } {TEXT 678 1 "j" }{TEXT -1 1 " " }{TEXT 671 1 "x" }{TEXT -1 1 " " } {TEXT 672 1 "i" }{TEXT -1 4 " )+ " }{XPPEDIT 18 0 "y[1]*y[2]" "6#*&&% \"yG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT -1 3 " ( " }{TEXT 679 1 "j" }{TEXT -1 1 " " }{TEXT 673 1 "x" }{TEXT -1 1 " " }{TEXT 674 1 "j" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 2 "+ " }{XPPEDIT 18 0 "y[1]*z[2] " "6#*&&%\"yG6#\"\"\"F'&%\"zG6#\"\"#F'" }{TEXT -1 3 " ( " }{TEXT 676 1 "j" }{TEXT -1 1 " " }{TEXT 675 1 "x" }{TEXT -1 1 " " }{TEXT 677 1 "k " }{TEXT -1 6 " ) + " }{XPPEDIT 18 0 "z[1]*x[2]" "6#*&&%\"zG6#\"\"\"F '&%\"xG6#\"\"#F'" }{TEXT -1 3 " ( " }{TEXT 689 1 "k" }{TEXT -1 1 " " } {TEXT 681 1 "x" }{TEXT -1 1 " " }{TEXT 682 1 "i" }{TEXT -1 5 " ) + " } {XPPEDIT 18 0 "z[1]*y[2]" "6#*&&%\"zG6#\"\"\"F'&%\"yG6#\"\"#F'" } {TEXT -1 3 " ( " }{TEXT 683 1 "k" }{TEXT -1 1 " " }{TEXT 684 1 "x" } {TEXT -1 1 " " }{TEXT 685 1 "j" }{TEXT -1 5 " ) + " }{XPPEDIT 18 0 "z[ 1]*z[2]" "6#*&&%\"zG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT -1 3 " ( " }{TEXT 686 1 "k" }{TEXT -1 1 " " }{TEXT 687 1 "x" }{TEXT -1 1 " " }{TEXT 688 1 "k" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Using the relations: " }{TEXT 713 1 "i" }{TEXT -1 1 " " }{TEXT 711 1 "x" }{TEXT -1 1 " " }{TEXT 712 1 "i" }{TEXT -1 5 " \+ = " }{TEXT 714 1 "j" }{TEXT -1 1 " " }{TEXT 715 1 "x" }{TEXT -1 1 " \+ " }{TEXT 716 1 "j" }{TEXT -1 5 " = " }{TEXT 717 1 "k" }{TEXT -1 1 " \+ " }{TEXT 718 1 "x" }{TEXT -1 1 " " }{TEXT 719 1 "k" }{TEXT -1 4 " = \+ " }{TEXT 720 1 "0" }{TEXT -1 3 ", " }{TEXT 693 1 "i" }{TEXT -1 1 " " }{TEXT 691 1 "x" }{TEXT -1 1 " " }{TEXT 692 1 "j" }{TEXT -1 5 " = " }{TEXT 705 1 "k" }{XPPEDIT 18 0 "``= -``" "6#/%!G,$F$!\"\"" }{TEXT -1 1 " " }{TEXT 694 1 "j" }{TEXT -1 1 " " }{TEXT 690 1 "x" }{TEXT -1 1 " \+ " }{TEXT 695 1 "i" }{TEXT -1 3 ", " }{TEXT 702 1 "j" }{TEXT -1 1 " " }{TEXT 697 1 "x" }{TEXT -1 1 " " }{TEXT 698 1 "k" }{TEXT -1 5 " = " }{TEXT 706 1 "i" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= -``" "6#/%!G,$F$! \"\"" }{TEXT -1 1 " " }{TEXT 703 1 "k" }{TEXT -1 1 " " }{TEXT 696 1 "x " }{TEXT -1 1 " " }{TEXT 704 1 "j" }{TEXT -1 3 ", " }{TEXT 707 1 "k" }{TEXT -1 1 " " }{TEXT 700 1 "x" }{TEXT -1 1 " " }{TEXT 708 1 "i" } {TEXT -1 5 " = " }{TEXT 709 1 "j" }{TEXT -1 2 " " }{XPPEDIT 18 0 "` `= -``" "6#/%!G,$F$!\"\"" }{TEXT -1 1 " " }{TEXT 710 1 "i" }{TEXT -1 1 " " }{TEXT 699 1 "x" }{TEXT -1 1 " " }{TEXT 701 1 "k" }{TEXT -1 7 " \+ gives:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 721 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 722 1 " " }{TEXT 724 1 " x" }{TEXT 723 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x[1]*y[2]" "6#*&&%\"xG6#\"\"\"F'&%\"yG6#\" \"#F'" }{TEXT -1 3 " ( " }{TEXT 725 1 "k" }{TEXT -1 5 " ) + " } {XPPEDIT 18 0 "x[1]*z[2]" "6#*&&%\"xG6#\"\"\"F'&%\"zG6#\"\"#F'" } {TEXT -1 2 " (" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 730 1 "j" }{TEXT -1 4 ") + " }{XPPEDIT 18 0 "y[1]*x[2]" "6#*&&%\"yG6# \"\"\"F'&%\"xG6#\"\"#F'" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "``-``" "6#,& %!G\"\"\"F$!\"\"" }{TEXT 731 1 "k" }{TEXT -1 5 " ) + " }{XPPEDIT 18 0 "y[1]*z[2]" "6#*&&%\"yG6#\"\"\"F'&%\"zG6#\"\"#F'" }{TEXT -1 3 " ( " } {TEXT 726 1 "i" }{TEXT -1 6 " ) + " }{XPPEDIT 18 0 "z[1]*x[2]" "6#*&& %\"zG6#\"\"\"F'&%\"xG6#\"\"#F'" }{TEXT -1 3 " ( " }{TEXT 727 1 "j" } {TEXT -1 5 " ) + " }{XPPEDIT 18 0 "z[1]*y[2]" "6#*&&%\"zG6#\"\"\"F'&% \"yG6#\"\"#F'" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "`` " "6#%!G" } {XPPEDIT 18 0 "-``" "6#,$%!G!\"\"" }{TEXT 728 1 "i" }{TEXT -1 4 " ). \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 "= \+ " }{XPPEDIT 18 0 "``(y[1]*z[2]-z[1]*y[2])" "6#-%!G6#,&*&&%\"yG6#\"\"\" F+&%\"zG6#\"\"#F+F+*&&F-6#F+F+&F)6#F/F+!\"\"" }{TEXT -1 2 " " }{TEXT 732 1 "i" }{TEXT -1 5 " + " }{XPPEDIT 18 0 "``(z[1]*x[2]-x[1]*z[2] ) " "6#-%!G6#,&*&&%\"zG6#\"\"\"F+&%\"xG6#\"\"#F+F+*&&F-6#F+F+&F)6#F/F+! \"\"" }{TEXT -1 2 " " }{TEXT 733 1 "j" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "``(x[1]*y[2]-y[1]*x[2]);" "6#-%!G6#,&*&&%\"xG6#\"\"\"F+&%\"yG6# \"\"#F+F+*&&F-6#F+F+&F)6#F/F+!\"\"" }{TEXT -1 1 " " }{TEXT 734 1 "k" } {TEXT -1 3 ". " }{XPPEDIT 18 0 "`` " "6#%!G" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 735 33 "_________________________________" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "Explanation of the distributive p roperty of the cross product " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 97 "In this section we give an exp lanation of the distributive law for the cross product of vectors: " } }{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 737 1 "u" }{TEXT -1 1 " " }{TEXT 736 1 "x" }{TEXT -1 3 " ( " }{TEXT 738 1 "v" }{TEXT -1 3 " + " }{TEXT 745 1 "w" }{TEXT -1 7 " ) = ( " }{TEXT 740 1 "u" }{TEXT -1 1 " \+ " }{TEXT 739 1 "x" }{TEXT -1 1 " " }{TEXT 741 1 "v" }{TEXT -1 7 " ) + \+ ( " }{TEXT 743 1 "u" }{TEXT -1 1 " " }{TEXT 742 1 "x" }{TEXT -1 1 " " }{TEXT 744 1 "w" }{TEXT -1 4 " ). " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 89 "We start by giving a geometrical inte rpretation of the construction of the cross product " }{TEXT 747 1 "u " }{TEXT -1 1 " " }{TEXT 746 1 "x" }{TEXT -1 1 " " }{TEXT 748 1 "v" } {TEXT -1 56 " which is slightly different from that given previously. " }}{PARA 257 "" 0 "" {TEXT -1 12 "The vectors " }{TEXT 750 1 "u" } {TEXT -1 5 " and " }{TEXT 751 1 "v" }{TEXT -1 128 " are drawn so that \+ they emanate from the common point O. A plane P is drawn through O so \+ that it is perpendicular to the vector " }{TEXT 749 1 "u" }{TEXT -1 18 " at O. The vector " }{TEXT 752 1 "v" }{TEXT -1 48 " is projected o nto the plane P to give a vector " }{TEXT 753 1 "v" }{TEXT -1 22 "', w hose length is || " }{TEXT 754 1 "v" }{TEXT -1 4 " || " }{XPPEDIT 18 0 "sin(theta)" "6#-%$sinG6#%&thetaG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 22 " is the angle between " } {TEXT 865 1 "u" }{TEXT -1 5 " and " }{TEXT 866 1 "v" }{TEXT -1 13 ". T he vector " }{TEXT 755 1 "v" }{TEXT -1 71 "' is then rotated about the point O in the plane P through an angle of " }{XPPEDIT 18 0 "90^o" "6 #)\"#!*%\"oG" }{TEXT -1 61 ". The direction of the rotation must be su ch that the triple " }{TEXT 756 1 "u" }{TEXT -1 2 ", " }{TEXT 757 1 "v " }{TEXT -1 3 "', " }{TEXT 758 1 "v" }{TEXT -1 96 "\" forms a right-ha nd triple. Note that the rotation is around the axis determined by the vector " }{TEXT 775 1 "u" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 20 "Finally, the vector " }{TEXT 762 1 "v" }{TEXT -1 44 "\" is mult iplied by the length of the vector " }{TEXT 763 1 "u" }{TEXT -1 12 " t o give || " }{TEXT 769 1 "u" }{TEXT -1 4 " || " }{TEXT 770 1 "v" } {TEXT -1 36 "\". This vector is the cross product " }{TEXT 760 1 "u" } {TEXT -1 1 " " }{TEXT 759 1 "x" }{TEXT -1 1 " " }{TEXT 761 1 "v" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{TEXT 765 1 "u " }{TEXT -1 1 " " }{TEXT 764 1 "x" }{TEXT -1 1 " " }{TEXT 766 1 "v" } {TEXT -1 6 " = || " }{TEXT 767 1 "u" }{TEXT -1 4 " || " }{TEXT 768 1 " v" }{TEXT -1 9 "\" and || " }{TEXT 861 1 "u" }{TEXT -1 1 " " }{TEXT 860 1 "x" }{TEXT -1 1 " " }{TEXT 862 1 "v" }{TEXT -1 9 " || = || " } {TEXT 863 1 "u" }{TEXT -1 7 " || || " }{TEXT 864 1 "v" }{TEXT -1 4 " | | " }{XPPEDIT 18 0 "sin(theta)" "6#-%$sinG6#%&thetaG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 734 321 321 {PLOTDATA 2 "6 B-%'CURVESG6$7%7$$\"\"!F)$\"\"#F)7$$\"#5F)F(7$$\"#9F)F*-%'COLOURG6&%$R GBGF)F)F)-%)POLYGONSG6%7&F'F,F/7$$\"\"%F)F;-%&COLORG6&F5$\"#$*!\"#F@F@ -%&STYLEG6#%,PATCHNOGRIDG-F76%7&7$$\"\"&F)$\"\"\"F)7$$\"\"*F)$\"\"$F)7 $FP$\"\"'F)7$FKF;-F>6&F5$\"#'*FBFZFZFC-F$6$7&FOFTFWFJ-F>6&F5$FQ!\"\"F[ oF[o-F$6&7$7$$\"\"(F)F*7$Fao$\"#XF\\o7%7$$\"++++]p!\"*$\"++++DVFjoFco7 $$\"++++]qFjoF[pFC-F36&F5$\"*++++\"!\")F(F(-F$6&7$F`o7$F-$F1F\\o7%7$$ \"+2e!)*z*Fjo$\"+M!H5\\\"FjoFhp7$$\"+$>%>!y*Fjo$\"+m4(HR\"FjoFC-F36&F5 F(F(Fbp-F$6&7$F`o7$$\"#bF\\o$\"#NF\\o7%7$$\"+hYk9cFjo$\"+hYk9LFjoFjq7$ $\"+R`N&o&Fjo$\"+R`N&Q$FjoFC-F>6&F5F)$\"#&)FBF)-F$6&7$F`o7$F[r$\"$D\"F B7%7$$\"+!ogBn&Fjo$\"+S'y-G\"FjoFas7$$\"+?$Rwi&Fjo$\"+g8sp8FjoFC-F>6&F 5F)$\"#vFBF)-F$6'7$7$Fao$\"$-#FB7$$\"#')F\\o$\"$q\"FB7%7$$\"+2e!)\\%)F jo$\"+M!H5y\"FjoFit7$$\"+$>%>I%)Fjo$\"+m4(Ho\"FjoFCF_t-%*THICKNESSG6#F +-F$6%7$7$$\"3++++++++b!#<$\"3+++++++]7Fbv7$F`v$\"3++++++++NFbv-F36&F5 $\")!\\DP\"FdpFjv$\")viobFdp-%*LINESTYLEGF[v-F$6$797$$\"3O+++]UzghFbv$ \"3-+++cVY#f\"Fbv7$$\"3L+++*4(QSiFbv$\"3'******>66Uc\"Fbv7$$\"3E+++,++ DjFbv$\"3(******\\[u#R:Fbv7$$\"3%**********))RT'Fbv$\"3#******pDWy^\"F bv7$$\"3e*****f$on1lFbv$\"31+++EN3+:Fbv7$$\"3,++++\"eBg'Fbv$\"33+++gu7 '[\"Fbv7$$\"3q*****pf/.q'Fbv$\"3++++vA3w9Fbv7$$\"39+++(*3x*z'Fbv$\"3)* *****4UC+Z\"Fbv7$$\"3U+++,+++pFbv$\"3)************zY\"Fbv7$$\"3%)***** \\5H-+(FbvFiy7$$\"3O+++1ap*4(Fbv$\"33+++wA3w9Fbv7$$\"31+++.>k(>(FbvF_y 7$$\"3S+++mJK$H(FbvFjx7$$\"37+++.5,'Q(Fbv$\"3++++eU%y^\"Fbv7$$\"3u**** *4++]Z(FbvF`x7$$\"3m*****H!HhfvFbv$\"3/+++86@k:Fbv7$$\"3j*****>v0#RwFb vFfw7$$\"3j******)zsJr(Fbv$\"36+++$>>Qi\"Fbv7$$\"3E+++66&4y(Fbv$\"3)** ****>*p.e;Fbv7$$\"3w*****>&[-UyFbv$\"35+++Ot&[p\"Fbv7$$\"3w*****\\@Hf* yFbv$\"32+++,++M6&F5$\"\")F\\oFc_lFc_l-F$6$7%F'F:F/ Fin-F$6$7.7$Fao$\"3#)*************z#Fbv7$$\"3#)*****Rb1x%pFbv$\"3=+++e d&pz#Fbv7$$\"3#)*****H46e*oFbv$\"3))*****>?Yyy#Fbv7$$\"38+++t&3Z%oFbv$ \"3%)*****4mSFx#Fbv7$$\"3g*****R\"zy%z'Fbv$\"3)******f4aCWIq#Fbv7$$\"3D+++3*)=/cF bv$\"37+++)z`@r#Fbv7$$\"3'******zU\"HbcFbv$\"3<+++R$fss#Fbv7$$\"3g**** *p37_q&Fbv$\"3-+++/fC[FFbv7$$\"33+++e4d`dFbv$\"3%)*****4x`\\x#Fbv7$$\" 3#)*************z&Fbv$\"3:+++x'zr!GFbv7$$\"3T+++ie9WeFbv$\"3/+++l$yY%G Fbv7$$\"3A+++mDn&)eFbv$\"3))*****fWkr)GFbv7$$\"3N+++pSECfFbv$\"3y***** fd9V$HFbv7$$\"37+++mmiffFbv$\"3%******H\"*pd)HFbv7$$\"3-+++F7\\\"*fFbv $\"3++++_)Q6/$FbvFhv-%%TEXTG6&7$$\"#sF\\o$\"#PF\\oQ\"u6\"F`p-%%FONTG6% %*HELVETICAG%%BOLDGF.-Fcgl6&7$$\"$X*FB$\"$t\"FBQ&u~~~vF[hlFeqF\\hl-Fcg l6&FchlQ\"xF[hlFeq-F]hl6$F_hlFd_l-Fcgl6&7$$\"#jF\\oFRQ\"vF[hlFjrF\\hl- Fcgl6&7$$\"#iF\\o$\"$&=FBQ#v'F[hlF_tF\\hl-Fcgl6&7$$\"#zF\\o$\"$3#FBQ#v \"F[hlF_tF\\hl-Fcgl6&7$$\"$-(FB$\"$J\"FBQ#90F[hlFhv-F]hl6$F_hlFQ-Fcgl6 &7$$\"$@(FBFipQ\"oF[hlFhvF\\il-Fcgl6&7$FM$\"$<#FBQ\"PF[hlF2F\\[m-Fcgl6 &7$$\"#()F\\oF[rQ#P'F[hlF2F\\[m-Fcgl6&7$FaoFiilQ\"OF[hlF2F\\[m-Fcgl6&7 $$\"#oF\\o$\"#DF\\oQ\"qF[hlF2-F]hl6$%'SYMBOLGF.-Fcgl6&7$$\"#dF\\o$\"$0 $FBF[]mF2F\\]m-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F[hlF]^m-F]hl6#% (DEFAULTG-%%VIEWG6$F`^mF`^m" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The function mapping \+ " }{TEXT 774 1 "v" }{TEXT -1 4 " to " }{TEXT 772 1 "u" }{TEXT -1 1 " \+ " }{TEXT 771 1 "x" }{TEXT -1 1 " " }{TEXT 773 1 "v" }{TEXT -1 57 " can be thought of as comprising of the three operations:" }}{PARA 15 "" 0 "" {TEXT -1 8 "Project " }{TEXT 778 1 "v" }{TEXT -1 33 " onto a plan e P perpendicular to " }{TEXT 779 1 "u" }{TEXT -1 2 ". " }}{PARA 15 " " 0 "" {TEXT -1 15 "Rotate through " }{XPPEDIT 18 0 "90^o" "6#)\"#!*% \"oG" }{TEXT -1 34 " in the plane P around the vector " }{TEXT 776 1 " u" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 26 "Multiply by the sc alar || " }{TEXT 777 1 "u" }{TEXT -1 4 " ||." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The distributive property follo ws because each of these operations \"" }{TEXT 259 25 "preserves vecto r addition" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 106 "To expl ain this in detail we investigate what happens when we apply the three operations to three vectors " }{TEXT 780 1 "v" }{TEXT -1 2 ", " } {TEXT 781 1 "w" }{TEXT -1 5 " and " }{TEXT 782 1 "v" }{TEXT -1 3 " + \+ " }{TEXT 783 1 "w" }{TEXT -1 69 " with reference to the parallelogram \+ law for the addition of vectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 "The three vectors " }{TEXT 785 1 "u" } {TEXT -1 2 ", " }{TEXT 786 1 "v" }{TEXT -1 5 " and " }{TEXT 787 1 "w" }{TEXT -1 58 " can be pictured as emanating from a single point O, wit h " }{TEXT 788 1 "v" }{TEXT -1 3 " + " }{TEXT 789 1 "w" }{TEXT -1 51 " forming the diagonal of a parallelogram which has " }{TEXT 790 1 "v" }{TEXT -1 5 " and " }{TEXT 791 1 "w" }{TEXT -1 22 " as two of its side s. " }}{PARA 0 "" 0 "" {TEXT -1 24 "After the three vectors " }{TEXT 792 1 "v" }{TEXT -1 2 ", " }{TEXT 793 1 "w" }{TEXT -1 5 " and " } {TEXT 794 1 "v" }{TEXT -1 3 " + " }{TEXT 795 1 "w" }{TEXT -1 40 " are \+ projected onto the plane P to give " }{TEXT 796 1 "v" }{TEXT -1 3 "', \+ " }{TEXT 797 1 "w" }{TEXT -1 7 "' and (" }{TEXT 798 1 "v" }{TEXT -1 3 " + " }{TEXT 799 1 "w" }{TEXT -1 16 ")', the vector (" }{TEXT 800 1 "v " }{TEXT -1 3 " + " }{TEXT 801 1 "w" }{TEXT -1 51 ")' forms the diagon al of a parallelogram which has " }{TEXT 802 1 "v" }{TEXT -1 6 "' and \+ " }{TEXT 803 1 "w" }{TEXT -1 29 "' as two of its sides. Thus (" } {TEXT 804 1 "v" }{TEXT -1 3 " + " }{TEXT 805 1 "w" }{TEXT -1 5 ")' = \+ " }{TEXT 806 1 "v" }{TEXT -1 4 "' + " }{TEXT 807 1 "w" }{TEXT -1 3 "'. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 643 398 398 {PLOTDATA 2 "6=-%'CURVESG6$7%7$$!\"\"\"\"!$F*F*7$$\"#6F*F+7$$\"#:F*$\" \")F*-%'COLOURG6&%$RGBGF*F*F*-%)POLYGONSG6%7&F'F,F/7$$\"\"$F*F2-%&COLO RG6&F7$\"#$*!\"#FBFB-%&STYLEG6#%,PATCHNOGRIDG-F$6$7%F'FFW*=#F]pFE-F@6&F7Fep$\"\"\"F)Fep-F$ 6&7$FS7$F\\rF\\r7%7$$\"*+++9(Fgn$\"++++qnF]pFbs7$$\"*+++E(Fgn$\"++++Ip F]pFE-F56&F7F_oF+F_o-F$6&7$FS7$F^rFT7%7$$\"*+++F#Fgn$\"+++++RF]pFct7$F ft$\"+++++TF]pFE-F56&F7$\")#)eqkFgn$\"))eqk\"FgnFau-F$6&7$FS7$F^r$\"#9 F*7%7$$\"*fqc>#Fgn$\"+`.Jj8FgnFfu7$$\"*THVM#Fgn$\"+Z'*ow8FgnFE-F@6&F7$ F3F)$F>F)Fgv-F$6%7%F[rFctFdo-F@6&F7$FUF)F]wF]w-%*LINESTYLEG6#F_r-F$6%7 %FbsFfuFjpF[wF^w-F$6%7$FctFfu-F@6&F7F]wF*$\"\"*F)F^w-F$6%7$FdoFjpFgwF^ w-F$6%7$F[rFbsFgwF^w-%%TEXTG6&7$$\"$8\"F)$\"$7\"F)Q\"u6\"F]o-%%FONTG6% %*HELVETICAG%%BOLDG\"#5-Fbx6&7$$\"#\"*F)$\"#tF)Q\"vFjxFfqF[y-Fbx6&7$$ \"#$)F)$\"#mF)Q\"wFjxF^tF[y-Fbx6&7$F\\r$\"#%*F)Q&v~+~wFjxFdvF[y-Fbx6&7 $$\"#\")F)$\"#dF)Q#v'FjxFcpF[y-Fbx6&7$$\"$X)FD$\"#LF)Q#w'FjxF[sF[y-Fbx 6&7$$\"#_F)$\"$b%FDQ3(v~+~w)'~=~v'~+~w'FjxF]uF[y-Fbx6&7$FguFfyQ\"PFjxF 4-F\\y6$F^yF`y-Fbx6&7$Fgx$\"#QF)Q\"OFjxF4Fc\\l-%*AXESSTYLEG6#%%NONEG-% +AXESLABELSG6%Q!FjxFb]l-F\\y6#%(DEFAULTG-%%VIEWG6$Fe]lFe]l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C urve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22 " "Curve 23" "Curve 24" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 24 "After the three vectors " }{TEXT 808 1 "v" }{TEXT -1 3 "', " }{TEXT 809 1 "w" }{TEXT -1 6 "' and " }{TEXT 810 1 "v" } {TEXT -1 4 "' + " }{TEXT 811 1 "w" }{TEXT -1 22 "' are rotated through " }{XPPEDIT 18 0 "90^o" "6#)\"#!*%\"oG" }{TEXT -1 34 " in the plane P around the vector " }{TEXT 784 1 "u" }{TEXT -1 10 ". to give " } {TEXT 812 1 "v" }{TEXT -1 3 "\", " }{TEXT 813 1 "w" }{TEXT -1 7 "\" an d (" }{TEXT 814 1 "v" }{TEXT -1 3 " + " }{TEXT 815 1 "w" }{TEXT -1 16 ")\", the vector (" }{TEXT 816 1 "v" }{TEXT -1 3 " + " }{TEXT 817 1 "w " }{TEXT -1 51 ")\" forms the diagonal of a parallelogram which has " }{TEXT 818 1 "v" }{TEXT -1 6 "\" and " }{TEXT 819 1 "w" }{TEXT -1 92 " \" as two of its sides. This operation does not change the shape of th e parallelogram. Thus (" }{TEXT 820 1 "v" }{TEXT -1 3 " + " }{TEXT 821 1 "w" }{TEXT -1 5 ")\" = " }{TEXT 822 1 "v" }{TEXT -1 4 "\" + " } {TEXT 823 1 "w" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 33 "Finally, when the three vectors " } {TEXT 824 1 "v" }{TEXT -1 3 "\", " }{TEXT 825 1 "w" }{TEXT -1 7 "\" an d (" }{TEXT 826 1 "v" }{TEXT -1 3 " + " }{TEXT 827 1 "w" }{TEXT -1 39 ")\" are all multiplied by the scalar || " }{TEXT 830 1 "u" }{TEXT -1 16 " ||, to give || " }{TEXT 836 1 "u" }{TEXT -1 4 " || " }{TEXT 837 1 "v" }{TEXT -1 6 "\", || " }{TEXT 838 1 "u" }{TEXT -1 4 " || " } {TEXT 839 1 "w" }{TEXT -1 9 "\" and || " }{TEXT 842 1 "u" }{TEXT -1 5 " || (" }{TEXT 840 1 "v" }{TEXT -1 3 " + " }{TEXT 841 1 "w" }{TEXT -1 18 ")\", the vector || " }{TEXT 831 1 "u" }{TEXT -1 5 " || (" }{TEXT 828 1 "v" }{TEXT -1 3 " + " }{TEXT 829 1 "w" }{TEXT -1 54 ")\" forms t he diagonal of a parallelogram which has || " }{TEXT 832 1 "u" }{TEXT -1 4 " || " }{TEXT 833 1 "v" }{TEXT -1 10 "\" and || " }{TEXT 834 1 " u" }{TEXT -1 4 " || " }{TEXT 835 1 "w" }{TEXT -1 109 "\" as two of its sides. This operation just changes the size of the parallelogram, and not its shape. Thus || " }{TEXT 845 1 "u" }{TEXT -1 5 " || (" }{TEXT 843 1 "v" }{TEXT -1 3 " + " }{TEXT 844 1 "w" }{TEXT -1 7 ")\" = | " } {TEXT 846 1 "u" }{TEXT -1 4 " || " }{TEXT 847 1 "v" }{TEXT -1 8 "\" + \+ || " }{TEXT 848 1 "u" }{TEXT -1 4 " || " }{TEXT 849 1 "w" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 30 "Altogether, this means that \+ " }{TEXT 851 1 "u" }{TEXT -1 1 " " }{TEXT 850 1 "x" }{TEXT -1 3 " ( " }{TEXT 852 1 "v" }{TEXT -1 3 " + " }{TEXT 859 1 "w" }{TEXT -1 7 " ) = \+ ( " }{TEXT 854 1 "u" }{TEXT -1 1 " " }{TEXT 853 1 "x" }{TEXT -1 1 " " }{TEXT 855 1 "v" }{TEXT -1 7 " ) + ( " }{TEXT 857 1 "u" }{TEXT -1 1 " \+ " }{TEXT 856 1 "x" }{TEXT -1 1 " " }{TEXT 858 1 "w" }{TEXT -1 5 " ). \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 102 "Deducing the geome trical description of the cross product from the description in terms \+ of components " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 143 "The geometrical description of the cross product can be deduced from the description in terms of components by first establishing the formula: " }}{PARA 256 "" 0 "" {TEXT -1 2 "( \+ " }{TEXT 945 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 946 1 " " }{TEXT 948 1 "x" }{TEXT 947 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&% !G6#\"\"#" }{TEXT -1 3 " ) " }{TEXT 953 1 "." }{TEXT -1 3 " ( " } {TEXT 949 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 950 1 " " }{TEXT 952 1 "x" }{TEXT 951 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6 #\"\"#" }{TEXT -1 9 " ) = ( " }{TEXT 954 1 "v" }{XPPEDIT 18 0 "``[1] " "6#&%!G6#\"\"\"" }{TEXT -1 3 " . " }{TEXT 955 1 "v" }{XPPEDIT 18 0 " ``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " ) " }{TEXT 960 1 "." }{TEXT -1 3 " ( " }{TEXT 956 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" } {TEXT -1 3 " . " }{TEXT 957 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\" \"#" }{TEXT -1 2 " )" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" } {TEXT -1 2 "( " }{TEXT 958 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\" \"" }{TEXT -1 1 " " }{TEXT 995 1 "." }{TEXT -1 1 " " }{TEXT 959 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 2 " )" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 12 " ------- (i)" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 961 37 "_____________________________________ " }{TEXT -1 17 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 12 "Explanation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 24 "The left side of (i) is " }} {PARA 256 "" 0 "" {TEXT -1 3 " ( " }{TEXT 962 1 "v" }{XPPEDIT 18 0 "`` [1]" "6#&%!G6#\"\"\"" }{TEXT 963 1 " " }{TEXT 965 1 "x" }{TEXT 964 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ) " }{TEXT 970 1 "." }{TEXT -1 3 " ( " }{TEXT 966 1 "v" }{XPPEDIT 18 0 "``[1]" "6 #&%!G6#\"\"\"" }{TEXT 967 1 " " }{TEXT 969 1 "x" }{TEXT 968 2 " v" } {XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 2 " )" }{XPPEDIT 18 0 "`` = (y[1]*z[2]-y[2]*z[1])^2+(x[2]*z[1]-x[1]*z[2])^2+(x[1]*y[2]-x[2 ]*y[1])^2;" "6#/%!G,(*$,&*&&%\"yG6#\"\"\"F,&%\"zG6#\"\"#F,F,*&&F*6#F0F ,&F.6#F,F,!\"\"F0F,*$,&*&&%\"xG6#F0F,&F.6#F,F,F,*&&F;6#F,F,&F.6#F0F,F6 F0F,*$,&*&&F;6#F,F,&F*6#F0F,F,*&&F;6#F0F,&F*6#F,F,F6F0F," }{TEXT -1 14 " ------- (ii)," }}{PARA 0 "" 0 "" {TEXT -1 30 "while the right sid e of (i) is" }}{PARA 0 "" 0 "" {TEXT -1 15 " ( " }{TEXT 971 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " . " } {TEXT 972 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " \+ ) " }{TEXT 977 1 "." }{TEXT -1 3 " ( " }{TEXT 973 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " . " }{TEXT 974 1 "v" } {XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 2 " )" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 2 "( " }{TEXT 975 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " . " }{TEXT 976 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 2 " )" } {XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 3 " = " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x[1]^2+y[1]^2+z[1]^2)*(x[2]^2+y[2 ]^2+z[2]^2) - (x[1]*x[2]+y[1]*y[2]+z[1]*z[2])^2" "6#,&*&,(*$&%\"xG6#\" \"\"\"\"#F**$&%\"yG6#F*F+F**$&%\"zG6#F*F+F*F*,(*$&F(6#F+F+F**$&F.6#F+F +F**$&F26#F+F+F*F*F**$,(*&&F(6#F*F*&F(6#F+F*F**&&F.6#F*F*&F.6#F+F*F**& &F26#F*F*&F26#F+F*F*F+!\"\"" }{TEXT -1 16 " ------- (iii). " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Expanding (ii ) gives . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "(y[1]*z[2]-y[2]*z[1])^2+(x[2]*z[1]-x[1]*z[2])^2 +\n (x[1]*y[2]-x[2]*y[1])^2;\ne1 := expand(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$),&*&&%\"yG6#\"\"\"F+&%\"zG6# \"\"#F+F+*&&F)F.F+&F-F*F+!\"\"F/F+F+*$),&*&&%\"xGF.F+F2F+F+*&&F9F*F+F, F+F3F/F+F+*$),&*&F;F+F1F+F+*&F8F+F(F+F3F/F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G,4*&)&%\"yG6#\"\"\"\"\"#F+)&%\"zG6#F,F,F+F+*,F,F+ F(F+F.F+&F)F0F+&F/F*F+!\"\"*&)F2F,F+)F3F,F+F+*&)&%\"xGF0F,F+F7F+F+*,F, F+F:F+F3F+&F;F*F+F.F+F4*&)F=F,F+F-F+F+*&F?F+F6F+F+*,F,F+F=F+F2F+F:F+F( F+F4*&F9F+F'F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Expanding (iii) gives . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "(x[1]^2+y[1]^2+z[1]^ 2)*(x[2]^2+y[2]^2+z[2]^2)-\n (x[1]*x[2]+y[1]*y[2]+z[1 ]*z[2])^2;\ne2 := expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,( *$)&%\"xG6#\"\"\"\"\"#F+F+*$)&%\"yGF*F,F+F+*$)&%\"zGF*F,F+F+F+,(*$)&F) 6#F,F,F+F+*$)&F0F9F,F+F+*$)&F4F9F,F+F+F+F+*$),(*&F(F+F8F+F+*&F/F+F%#e2G,4*&)& %\"yG6#\"\"\"\"\"#F+)&%\"zG6#F,F,F+F+*,F,F+F(F+F.F+&F)F0F+&F/F*F+!\"\" *&)F2F,F+)F3F,F+F+*&)&%\"xGF0F,F+F7F+F+*,F,F+F:F+F3F+&F;F*F+F.F+F4*&)F =F,F+F-F+F+*&F?F+F6F+F+*,F,F+F=F+F2F+F:F+F(F+F4*&F9F+F'F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "These two expre ssions are the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "is(e1=e2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "Since " }{TEXT 988 1 "v" }{XPPEDIT 18 0 "``[1]" "6# &%!G6#\"\"\"" }{TEXT -1 1 " " }{TEXT 994 1 "." }{TEXT -1 1 " " }{TEXT 989 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 6 " = || \+ " }{TEXT 990 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 7 " || || " }{TEXT 991 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" } {TEXT -1 4 " || " }{XPPEDIT 18 0 "cos(theta)" "6#-%$cosG6#%&thetaG" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 22 " is the angle between " }{TEXT 992 1 "v" }{XPPEDIT 18 0 "``[1]" "6 #&%!G6#\"\"\"" }{TEXT -1 3 " . " }{TEXT 993 1 "v" }{XPPEDIT 18 0 "``[2 ];" "6#&%!G6#\"\"#" }{TEXT -1 10 ", and || " }{TEXT 978 1 "u" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 3 " = " } {TEXT 980 1 "u" }{TEXT -1 1 " " }{TEXT 979 1 "." }{TEXT -1 1 " " } {TEXT 981 1 "u" }{TEXT -1 17 ", for any vector " }{TEXT 996 1 "u" } {TEXT -1 34 ", (i) can be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 4 " || " }{TEXT 982 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\" \"\"" }{TEXT 983 1 " " }{TEXT 985 1 "x" }{TEXT 984 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 6 " = || " }{TEXT 986 1 "v" }{XPPEDIT 18 0 "` `[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%! G\"\"#" }{TEXT -1 4 " || " }{TEXT 987 1 "v" }{XPPEDIT 18 0 "``[2];" "6 #&%!G6#\"\"#" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" } {XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 3 "|| " }{TEXT 997 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " ||" } {XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 4 " || " }{TEXT 998 1 "v " }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(theta)^ 2" "6#*$-%$cosG6#%&thetaG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 6 " || " }{TEXT 999 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 1000 1 " " } {TEXT 1002 1 "x" }{TEXT 1001 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6# \"\"#" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 6 " = || " }{TEXT 1003 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\" " }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 4 " || " }{TEXT 1004 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" } {TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2*``(1-cos(theta)^2);" "6#*&%!G\" \"#-F$6#,&\"\"\"F)*$-%$cosG6#%&thetaGF%!\"\"F)" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 7 " = || " }{TEXT 1005 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6# *$%!G\"\"#" }{TEXT -1 4 " || " }{TEXT 1006 1 "v" }{XPPEDIT 18 0 "``[2] ;" "6#&%!G6#\"\"#" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2*sin(theta)^2 ;" "6#*&%!G\"\"#-%$sinG6#%&thetaGF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 4 " or " }}{PARA 256 "" 0 "" {TEXT -1 7 " || " }{TEXT 1007 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 1008 1 " " } {TEXT 1010 1 "x" }{TEXT 1009 2 " v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6# \"\"#" }{TEXT -1 10 " || = || " }{TEXT 1011 1 "v" }{XPPEDIT 18 0 "``[ 1]" "6#&%!G6#\"\"\"" }{TEXT -1 8 " || || " }{TEXT 1012 1 "v" } {XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 5 " || " }{XPPEDIT 18 0 "sin(theta);" "6#-%$sinG6#%&thetaG" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 20 "( Note that, since " }{XPPEDIT 18 0 "0<=theta" "6 #1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "sin(theta)>=0" "6#1\"\"!-%$sinG6#%&thetaG" }{TEXT -1 4 ". ) " }}{PARA 0 "" 0 "" {TEXT -1 42 "This formula, together with the fact that " }{TEXT 1013 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\" \"\"" }{TEXT 1014 1 " " }{TEXT 1016 1 "x" }{TEXT 1015 2 " v" } {XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 23 " is orthogonal to both " }{TEXT 1018 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 5 " and " }{TEXT 1017 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6# \"\"#" }{TEXT -1 37 ", gives the geometrical description. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exam ple 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 882 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 50 "Find a unit vector which is perpendicular to both " }{TEXT 884 1 " u" }{TEXT -1 3 " = " }{TEXT 885 1 "i" }{XPPEDIT 18 0 "``- 2" "6#,&%!G \"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 886 1 "j" }{TEXT -1 5 " + 3 \+ " }{TEXT 887 1 "k" }{TEXT -1 5 " and " }{TEXT 888 1 "v" }{TEXT -1 5 " \+ = 3 " }{TEXT 889 1 "i" }{TEXT -1 6 " + 2 " }{TEXT 890 1 "j" } {XPPEDIT 18 0 "``-4;" "6#,&%!G\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " } {TEXT 891 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 883 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 18 "The cross product " }{TEXT 893 1 "u" }{TEXT -1 1 " \+ " }{TEXT 892 1 "x" }{TEXT -1 1 " " }{TEXT 894 1 "v" }{TEXT -1 26 " is \+ perpendicular to both " }{TEXT 895 1 "u" }{TEXT -1 5 " and " }{TEXT 896 1 "v" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 906 2 "u " }{TEXT 907 1 "x" }{TEXT 908 2 " v" }{TEXT -1 7 " = det " } {XPPEDIT 18 0 "matrix([[\"i\", \"j\", \"k\"], [1, -2, 3], [3, 2, -4]]) ;" "6#-%'matrixG6#7%7%Q\"i6\"Q\"jF)Q\"kF)7%\"\"\",$\"\"#!\"\"\"\"$7%F1 F/,$\"\"%F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 " = det " }{XPPEDIT 18 0 "matrix([[-2, 3], [2, -4]]);" "6#-%'matrixG6#7$7$,$\"\"#!\"\"\"\"$7$F),$\"\"%F*" }{TEXT -1 1 " " }{TEXT 900 1 "i" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" } {TEXT -1 4 "det " }{XPPEDIT 18 0 "matrix([[1, 3], [3, -4]]);" "6#-%'ma trixG6#7$7$\"\"\"\"\"$7$F),$\"\"%!\"\"" }{TEXT -1 1 " " }{TEXT 901 1 " j" }{TEXT -1 7 " + det " }{XPPEDIT 18 0 "matrix([[1, -2], [3, 2]]);" " 6#-%'matrixG6#7$7$\"\"\",$\"\"#!\"\"7$\"\"$F*" }{TEXT -1 1 " " }{TEXT 902 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(8-6);" "6#/%!G-F$6#,&\" \")\"\"\"\"\"'!\"\"" }{TEXT -1 1 " " }{TEXT 897 1 "i" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "-``(-4-9);" "6#,$-%!G6#,&\"\"%!\"\"\"\"*F)F)" } {TEXT -1 1 " " }{TEXT 898 1 "j" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "``( 2+6);" "6#-%!G6#,&\"\"#\"\"\"\"\"'F(" }{TEXT -1 1 " " }{TEXT 899 1 "k " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 903 2 " " }{TEXT -1 3 "= 2" }{TEXT 910 4 " i \+ " }{TEXT -1 5 "+ 13 " }{TEXT 904 1 "j" }{TEXT -1 6 " + 8 " }{TEXT 905 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 14 "The length of " }{TEXT 925 1 "u" }{TEXT -1 1 " " } {TEXT 924 1 "x" }{TEXT -1 1 " " }{TEXT 926 1 "v" }{TEXT -1 7 " is || \+ " }{TEXT 913 1 "u" }{TEXT -1 1 " " }{TEXT 912 1 "x" }{TEXT -1 1 " " } {TEXT 914 1 "v" }{TEXT -1 6 " || = " }{XPPEDIT 18 0 "sqrt(4+169+64) = \+ sqrt(237)" "6#/-%%sqrtG6#,(\"\"%\"\"\"\"$p\"F)\"#kF)-F%6#\"$P#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " || " }{TEXT 919 1 "u" }{TEXT -1 1 " " }{TEXT 918 1 "x" } {TEXT -1 1 " " }{TEXT 920 1 "v" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^( -1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 3 " ( " }{TEXT 922 1 "u" }{TEXT -1 1 " " }{TEXT 921 1 "x" }{TEXT -1 1 " " }{TEXT 923 1 "v" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "1/sqrt(237)" "6#*&\"\"\"F$-%%sqrtG6#\"$P#!\" \"" }{TEXT -1 4 " ( 2" }{TEXT 929 4 " i " }{TEXT -1 5 "+ 13 " }{TEXT 927 1 "j" }{TEXT -1 6 " + 8 " }{TEXT 928 1 "k" }{TEXT -1 5 " ) = " } {XPPEDIT 18 0 "2/sqrt(237)" "6#*&\"\"#\"\"\"-%%sqrtG6#\"$P#!\"\"" } {TEXT 917 4 " i " }{TEXT -1 2 "+ " }{XPPEDIT 18 0 "13/sqrt(237)" "6#* &\"#8\"\"\"-%%sqrtG6#\"$P#!\"\"" }{TEXT -1 1 " " }{TEXT 915 1 "j" } {TEXT -1 4 " + " }{XPPEDIT 18 0 "8/sqrt(237)" "6#*&\"\")\"\"\"-%%sqrt G6#\"$P#!\"\"" }{TEXT -1 1 " " }{TEXT 916 1 "k" }{TEXT -1 50 " is a u nit vector which is perpendicular to both " }{TEXT 909 1 "u" }{TEXT -1 5 " and " }{TEXT 911 1 "v" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "with(linalg):\nu : = [1,-2,3];\nv := [3,2,-4];\nw := crossprod(u,v);\nscalarmul(w,norm(w, 2)^(-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"\"!\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%\"\"$\"\"#!\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"wG-%'vectorG6#7%\"\"#\"#8\"\")" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$*$-%%sqrtG6#\"$P#\"\"\"#\"\"#F,, $F(#\"#8F,,$F(#\"\")F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT 867 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 44 "Find the area of the triangle with vertices " } {XPPEDIT 18 0 "A(3,0,-1)" "6#-%\"AG6%\"\"$\"\"!,$\"\"\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B(4,2,5)" "6#-%\"BG6%\"\"%\"\"#\"\"&" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "C(7,-2,4)" "6#-%\"CG6%\"\"(,$\"\"#! \"\"\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 868 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT 936 2 "AB" }{TEXT -1 4 " = " }{TEXT 930 1 "i" }{TEXT -1 5 " + 2 " }{TEXT 931 1 "j" }{TEXT -1 5 " + 6 " }{TEXT 932 1 "k" }{TEXT -1 5 " and " }{TEXT 937 2 "AC" }{TEXT -1 5 " = 4 " }{TEXT 933 1 "i" } {XPPEDIT 18 0 "``-2" "6#,&%!G\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 934 1 "j" }{TEXT -1 5 " + 5 " }{TEXT 935 1 "k" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 33 "The area of the triangle ABC is " } {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 4 " || " } {TEXT 938 2 "AB" }{TEXT -1 1 " " }{TEXT 940 1 "x" }{TEXT -1 1 " " } {TEXT 939 2 "AC" }{TEXT -1 5 " ||. " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 878 3 "AB " }{TEXT 879 1 "x" }{TEXT 880 3 " AC" }{TEXT -1 7 " = det " }{XPPEDIT 18 0 "matrix([[\"i\", \"j\", \"k\"], [1, 2, 6], [4, -2, 5]]);" "6#-%'matrixG6#7%7%Q\"i6\"Q\"jF)Q\"kF)7%\"\"\"\"\"#\"\"'7% \"\"%,$F.!\"\"\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 8 " = det " }{XPPEDIT 18 0 "matrix([[2, 6 ], [-2, 5]]);" "6#-%'matrixG6#7$7$\"\"#\"\"'7$,$F(!\"\"\"\"&" }{TEXT -1 1 " " }{TEXT 872 1 "i" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\" " }{TEXT -1 4 "det " }{XPPEDIT 18 0 "matrix([[1, 6], [4, 5]]);" "6#-%' matrixG6#7$7$\"\"\"\"\"'7$\"\"%\"\"&" }{TEXT -1 1 " " }{TEXT 873 1 "j " }{TEXT -1 7 " + det " }{XPPEDIT 18 0 "matrix([[1, 2], [4, -2]]);" "6 #-%'matrixG6#7$7$\"\"\"\"\"#7$\"\"%,$F)!\"\"" }{TEXT -1 1 " " }{TEXT 874 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(10+12);" "6#/%!G-F$6#,& \"#5\"\"\"\"#7F)" }{TEXT -1 1 " " }{TEXT 869 1 "i" }{TEXT -1 2 " " } {XPPEDIT 18 0 "-``(5-24);" "6#,$-%!G6#,&\"\"&\"\"\"\"#C!\"\"F+" } {TEXT -1 1 " " }{TEXT 870 1 "j" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "``( -2-8);" "6#-%!G6#,&\"\"#!\"\"\"\")F(" }{TEXT -1 1 " " }{TEXT 871 1 "k " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 875 2 " " }{TEXT -1 4 "= 22" }{TEXT 881 4 " i \+ " }{TEXT -1 5 "+ 19 " }{TEXT 876 1 "j" }{XPPEDIT 18 0 " ``-10" "6#,&%! G\"\"\"\"#5!\"\"" }{TEXT -1 1 " " }{TEXT 877 1 "k" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The area \+ of triangle ABC is " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 4 " || " }{TEXT 941 2 "AB" }{TEXT -1 1 " " }{TEXT 943 1 "x" } {TEXT -1 1 " " }{TEXT 942 2 "AC" }{TEXT -1 6 " || = " }{XPPEDIT 18 0 " sqrt(22^2+19^2+10^2)/2 = 3/2*sqrt(105)" "6#/*&-%%sqrtG6#,(*$\"#A\"\"# \"\"\"*$\"#>F+F,*$\"#5F+F,F,F+!\"\"*(\"\"$F,F+F1-F&6#\"$0\"F," }{TEXT -1 1 " " }{TEXT 944 1 "~" }{TEXT -1 14 " 15.37042616. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "with(li nalg):\nu := [1,2,6];\nv := [4,-2,5];\ncrossprod(u,v);\nnorm(%,2)/2;\n evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"\"\"\"#\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%\"\"%!\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"#A\"#>!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*$-%%sqrtG6#\"$0\"\"\"\"#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+;E/P:!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 50 "Find a unit vector which is perpendicular to both " }{TEXT 1020 1 "u" }{XPPEDIT 18 0 "`` = 4" "6#/%!G\"\"%" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``+``;" "6#,&%!G\"\"\"F$F%" }{TEXT -1 1 " " }{TEXT 1021 1 "j" }{XPPEDIT 18 0 " ``-5" "6#,&%!G\"\"\"\"\"&!\" \"" }{TEXT -1 1 " " }{TEXT 1022 1 "k" }{TEXT -1 5 " and " }{TEXT 1023 1 "v" }{XPPEDIT 18 0 "``" "6#%!G" }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\"\" #" }{TEXT -1 1 " " }{TEXT 1024 1 "i" }{XPPEDIT 18 0 "`` + 3" "6#,&%!G \"\"\"\"\"$F%" }{TEXT -1 1 " " }{TEXT 1025 1 "j" }{XPPEDIT 18 0 "``-`` ;" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 1026 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 50 "Find a unit vector which is perpendi cular to both " }{TEXT 1019 1 "u" }{XPPEDIT 18 0 "`` = 4" "6#/%!G\"\"% " }{TEXT -1 1 " " }{TEXT 1027 1 "i" }{XPPEDIT 18 0 "``+2;" "6#,&%!G\" \"\"\"\"#F%" }{TEXT -1 1 " " }{TEXT 1028 1 "j" }{XPPEDIT 18 0 "``-2;" "6#,&%!G\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 1029 1 "k" }{TEXT -1 5 " and " }{TEXT 1030 1 "v" }{XPPEDIT 18 0 "``" "6#%!G" }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\"\"#" }{TEXT -1 1 " " }{TEXT 1031 1 "i" }{XPPEDIT 18 0 "``-5;" "6#,&%!G\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 1032 1 " j" }{XPPEDIT 18 0 "``+5;" "6#,&%!G\"\"\"\"\"&F%" }{TEXT -1 1 " " } {TEXT 1033 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "_____ ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________ _______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 1046 1 "u" }{TEXT -1 3 " = " }{TEXT 1047 1 "i " }{TEXT -1 2 ", " }{TEXT 1048 1 "v" }{TEXT -1 3 " = " }{TEXT 1049 1 " i" }{TEXT -1 3 " + " }{TEXT 1050 1 "j" }{TEXT -1 7 " and " }{TEXT 1051 1 "w" }{TEXT -1 3 " = " }{TEXT 1052 1 "i" }{TEXT -1 3 " + " } {TEXT 1053 1 "j" }{TEXT -1 3 " + " }{TEXT 1054 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "Prove that the vector product is " } {TEXT 259 15 "not associative" }{TEXT -1 29 " by calculating and compa ring" }{TEXT 1040 2 " u" }{TEXT -1 1 " " }{TEXT 1038 1 "x" }{TEXT -1 3 " ( " }{TEXT 1041 1 "v" }{TEXT -1 1 " " }{TEXT 1037 1 "x" }{TEXT -1 1 " " }{TEXT 1042 1 "w" }{TEXT -1 9 " ) and ( " }{TEXT 1035 1 "u" } {TEXT -1 1 " " }{TEXT 1034 1 "x" }{TEXT -1 1 " " }{TEXT 1036 1 "v" } {TEXT -1 3 " ) " }{TEXT 1039 1 "x" }{TEXT -1 1 " " }{TEXT 1043 1 "w" } {TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "_____________________ ______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 44 "Find the are a of the triangle with vertices " }{XPPEDIT 18 0 "A(1, 3, -2);" "6#-% \"AG6%\"\"\"\"\"$,$\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B(2, 4 , 5);" "6#-%\"BG6%\"\"#\"\"%\"\"&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C(-3,-2,2);" "6#-%\"CG6%,$\"\"$!\"\",$\"\"#F(F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Ans. " }{XPPEDIT 18 0 "sqrt(2546)/2" "6#*& -%%sqrtG6#\"%YD\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 1044 1 "~" } {TEXT -1 9 " 25.229. " }}{PARA 0 "" 0 "" {TEXT -1 35 "________________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 49 "This questio n is concerned with the three points " }{XPPEDIT 18 0 "A(2, -2, 1);" " 6#-%\"AG6%\"\"#,$F&!\"\"\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B(3, \+ -1, 2);" "6#-%\"BG6%\"\"$,$\"\"\"!\"\"\"\"#" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "C(3,-1,1);" "6#-%\"CG6%\"\"$,$\"\"\"!\"\"F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "(a) Find a vector " }{TEXT 1059 1 "n" }{TEXT -1 71 " which is perpendicular to the plane containi ng the points A, B and C. " }}{PARA 0 "" 0 "" {TEXT -1 39 "(b) Find th e area of the triangle ABC. " }}{PARA 0 "" 0 "" {TEXT -1 74 "(c) Find \+ a unit vector perpendicular to the plane containing A, B and C. " }} {PARA 0 "" 0 "" {TEXT -1 10 "Ans. (a) " }{TEXT 1060 1 "i" }{XPPEDIT 18 0 " ``-`` " "6#,&%!G\"\"\"F$!\"\"" }{TEXT 1061 1 "j" }{TEXT -1 8 ", (b) " }{XPPEDIT 18 0 "sqrt(2)/2" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\" " }{TEXT -1 8 ", (c) " }{XPPEDIT 18 0 "1/sqrt(2)" "6#*&\"\"\"F$-%%sq rtG6#\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 1062 1 "i" }{XPPEDIT 18 0 "`` \+ - 1/sqrt(2)" "6#,&%!G\"\"\"*&F%F%-%%sqrtG6#\"\"#!\"\"F+" }{TEXT -1 1 " " }{TEXT 1063 1 "j" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "_ __________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________ _______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 49 "This question is concerned with the three points " } {XPPEDIT 18 0 "A(1,1,1);" "6#-%\"AG6%\"\"\"F&F&" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "B(2, 3, 4);" "6#-%\"BG6%\"\"#\"\"$\"\"%" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "C(3,0,-1);" "6#-%\"CG6%\"\"$\"\"!,$\"\"\"!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "(a) Find a vector " } {TEXT 1045 1 "n" }{TEXT -1 71 " which is perpendicular to the plane co ntaining the points A, B and C. " }}{PARA 0 "" 0 "" {TEXT -1 39 "(b) F ind the area of the triangle ABC. " }}{PARA 0 "" 0 "" {TEXT -1 74 "(c) Find a unit vector perpendicular to the plane containing A, B and C. \+ " }}{PARA 0 "" 0 "" {TEXT -1 10 "Ans. (a) " }{XPPEDIT 18 0 "``- ``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 1055 1 "i" }{XPPEDIT 18 0 "``+8;" "6#,&% !G\"\"\"\"\")F%" }{TEXT -1 1 " " }{TEXT 1064 1 "j" }{XPPEDIT 18 0 " `` -5" "6#,&%!G\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 1056 1 "k" } {TEXT -1 8 ", (b) " }{XPPEDIT 18 0 "3*sqrt(10)/2;" "6#*(\"\"$\"\"\"- %%sqrtG6#\"#5F%\"\"#!\"\"" }{TEXT -1 8 ", (c) " }{XPPEDIT 18 0 "-1/( 3*sqrt(10));" "6#,$*&\"\"\"F%*&\"\"$F%-%%sqrtG6#\"#5F%!\"\"F," }{TEXT -1 1 " " }{TEXT 1057 1 "i" }{XPPEDIT 18 0 "``+8/(3*sqrt(10));" "6#,&%! G\"\"\"*&\"\")F%*&\"\"$F%-%%sqrtG6#\"#5F%!\"\"F%" }{TEXT -1 1 " " } {TEXT 1058 1 "j" }{XPPEDIT 18 0 " ``- 5/(3*sqrt(10))" "6#,&%!G\"\"\"*& \"\"&F%*&\"\"$F%-%%sqrtG6#\"#5F%!\"\"F." }{TEXT -1 1 " " }{TEXT 1065 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "________________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 37 "geometrical picture \+ of cross product " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1396 "d := evalf(Pi/36):\np1 := plottools[arrow]( [0,0],[3,1],.1,.1,.03,\n arrow,color=red):\np2 := plottoo ls[arrow]([0,0],[5,0],.1,.1,.02,\n arrow,color=COLOR(RGB, 0,.7,0)):\np3 := plottools[arrow]([0,0],[0,2.2],.1,.1,.03,\n \+ arrow,color=blue,thickness=2):\np4 := plots[polygonplot]([[0,0],[3 ,1],[5,0]],\n color=COLOR(RGB,.96,.96,.96),style=patchnogrid):\np 5 := plot([[3,1],[5,0]],linestyle=2,color=grey):\np6 := plot([seq([cos (i*d),0.4*sin(i*d)],i=0..8)],color=black):\np7 := plot([seq([cos(i*d), 1.35+0.4*sin(i*d)],i=-12..15)],color=navy):\np8 := plot([[0.329,1.686] ,[0.259,1.736],[0.329,1.776]],color=navy):\nt1 := plots[textplot]([.7, .13,`q`],color=black,font=[SYMBOL,10]):\nt2 := plots[textplot]([1.6,.7 5,`v`],color=red,font=[HELVETICA,BOLD,10]):\nt3 := plots[textplot]([1. 7,.69,`2`],color=red,font=[HELVETICA,8]):\nt4 := plots[textplot]([2.6, -.07,`v`],color=COLOR(RGB,0,.7,0),\n font=[HELVETI CA,BOLD,10]):\nt5 := plots[textplot]([2.68,-.13,`1`],color=COLOR(RGB,0 ,.7,0),\n font=[HELVETICA,8]):\nt6 := plots[textpl ot]([-.35,1.35,`v v`],\n color=blue,font=[HELVETICA,BOLD ,10]):\nt7 := plots[textplot]([-.28,1.29,`1 2`],color=blue,font= [HELVETICA,8]):\nt8 := plots[textplot]([-.31,1.33,`x`],color=blue,font =[HELVETICA,8]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,\n \+ t1,t2,t3,t4,t5,t6,t7,t8],axes=none);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 13 "i-j-k cycle " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 491 "d := evalf(Pi/36) :\ncs := evalf(sqrt(3)/2):\np1 := plot([[seq([cos(i*d),sin(i*d)],i=-4. .16)],\n [seq([cos(i*d),sin(i*d)],i=20..40)],\n [seq([cos(i*d) ,sin(i*d)],i=44..64)]],color=red):\np2 := plot([[[1.01,-.3],[0.9397,-. 342],[0.9,-.25]],\n [[-.77,-.73],[-.766,-.643],[-0.69,-.62]],\n \+ [[-.23,.91],[-.174,.985],[-.25,1.04]]],color=red):\nt1 := plots[te xtplot]([[0,1,`i`],[cs,-.5,`j`],[-cs,-.5,`k`]],\n color=black,f ont=[HELVETICA,BOLD,12]):\nplots[display]([p1,p2,t1],axes=none);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 55 "alternative geometric d escription of the cross product " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2145 "d := evalf(Pi/36):\np1 := \+ plot([[0,2],[10,0],[14,2]],color=black):\np2 := plots[polygonplot]([[0 ,2],[10,0],[14,2],[4,4]],\n color=COLOR(RGB,.93,.93,.93),style=pa tchnogrid):\np3 := plots[polygonplot]([[5,1],[9,3],[9,6],[5,4]],\n \+ color=COLOR(RGB,.96,.96,.96),style=patchnogrid):\np4 := plot([[9,3], [9,6],[5,4],[5,1]],color=COLOR(RGB,.9,.9,.9)):\np5 := plottools[arrow] ([7,2],[7,4.5],.1,.1,.07,arrow,color=red):\np6 := plottools[arrow]([7, 2],[10,1.4],.1,.1,.07,arrow,color=blue):\np7 := plottools[arrow]([7,2] ,[5.5,3.5],.1,.1,.1,arrow,\n color=COLOR(RGB,0,.85,0)): \np8 := plottools[arrow]([7,2],[5.5,1.25],.1,.1,.1,arrow,\n \+ color=COLOR(RGB,0,.75,0)):\np9 := plottools[arrow]([7,2.02],[8.6,1 .7],.1,.1,.1,arrow,\n color=COLOR(RGB,0,.75,0),thickness =2):\np10 := plot([[5.5,1.25],[5.5,3.5]],color=navy,linestyle=2):\np11 := plot([seq([6.9+1.15*cos(i*d),2+0.532*sin(i*d)],i=46..68)],color=na vy):\np12 := plot([[7.81,1.76],[7.96,1.82],[7.99,1.71]],color=navy):\n p13 := plot([[5,1],[9,3]],color=COLOR(RGB,.8,.8,.8)):\np14 := plot([[0 ,2],[4,4],[14,2]],color=COLOR(RGB,.9,.9,.9)):\np15 := plot([[seq([7+.6 *cos(i*d),2+.8*sin(i*d)],i=18..29)],\n [seq([5.5+.6*cos(i*d), 3.5+.8*sin(i*d)],i=54..65)]],color=navy):\nt1 := plots[textplot]([7.2, 3.7,`u`],color=red,font=[HELVETICA,BOLD,10]):\nt2 := plots[textplot]([ 9.45,1.73,`u v`],color=blue,font=[HELVETICA,BOLD,10]):\nt3 := plots[ textplot]([9.45,1.73,`x`],color=blue,font=[HELVETICA,8]):\nt4 := plots [textplot]([6.3,3,`v`],color=COLOR(RGB,0,.85,0),\n font=[HEL VETICA,BOLD,10]):\nt5 := plots[textplot]([[6.2,1.85,`v'`],[7.9,2.08,`v \"`]],\n color=COLOR(RGB,0,.75,0),font=[HELVETICA,BOLD,10]): \nt6 := plots[textplot]([7.02,1.31,`90`],color=navy,font=[HELVETICA,9] ):\nt7 := plots[textplot]([7.21,1.4,`o`],color=navy,font=[HELVETICA,8] ):\nt8 := plots[textplot]([[1,2.17,`P`],[8.7,5.5,`P'`],\n \+ [7,1.85,`O`]],color=black,font=[HELVETICA,9]):\nt9 := plots[te xtplot]([[6.8,2.5,`q`],[5.7,3.05,`q`]],\n color=black,fon t=[SYMBOL,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p1 2,p13,p14,p15,\n t1,t2,t3,t4,t5,t6,t7,t8,t9],axes=none);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "projection of parallelogram \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2161 "d := evalf(Pi/36):\np1 := plot([[-1,0],[11,0],[15,8]],color= black):\np2 := plots[polygonplot]([[-1,0],[11,0],[15,8],[3,8]],\n \+ color=COLOR(RGB,.93,.93,.93),style=patchnogrid):\np3 := plot([[-1,0], [3,8],[15,8]],color=COLOR(RGB,.85,.85,.85)):\np4 := plottools[arrow]([ 11,4],[11,16],.1,.15,.04,arrow,color=red):\np5 := plottools[arrow]([11 ,4],[6,6],.1,.2,.05,arrow,\n color=COLOR(RGB ,0,.75,0)):\np6 := plottools[arrow]([11,4],[6,11],.1,.2,.04,arrow,\n \+ color=COLOR(RGB,0,.85,0)):\np7 := plottools[a rrow]([11,4],[7,2],.1,.2,.05,arrow,\n color= COLOR(RGB,.75,.1,.75)):\np8 := plottools[arrow]([11,4],[7,7],.1,.2,.05 ,arrow,\n color=magenta):\np9 := plottools[a rrow]([11,4],[2,4],.1,.2,.03,arrow,\n color= brown):\np10 := plottools[arrow]([11,4],[2,14],.1,.2,.03,arrow,\n \+ color=COLOR(RGB,.8,.3,.3)):\np11:= plot([[[7,2], [2,4],[6,6]],[[7,7],[2,14],[6,11]]],\n color=COLOR(RGB,.4,.4,. 4),linestyle=2):\np12 := plot([[[2,4],[2,14]],[[6,6],[6,11]],[[7,2],[7 ,7]]],\n color=COLOR(RGB,.4,0,.9),linestyle=2):\np23 := plot([ [5,1],[9,3]],color=COLOR(RGB,.8,.8,.8)):\np24 := plot([[0,2],[4,4],[14 ,2]],color=COLOR(RGB,.9,.9,.9)):\nt1 := plots[textplot]([11.3,11.2,`u` ],color=red,font=[HELVETICA,BOLD,10]):\nt2 := plots[textplot]([9.1,7.3 ,`v`],\n color=COLOR(RGB,0,.85,0),font=[HELVETICA,BOLD ,10]):\nt3 := plots[textplot]([8.3,6.6,`w`],\n color=m agenta,font=[HELVETICA,BOLD,10]):\nt4 := plots[textplot]([7,9.4,`v + w `],\n color=COLOR(RGB,.8,.3,.3),font=[HELVETICA,BOLD,1 0]):\nt5 := plots[textplot]([8.1,5.7,`v'`],\n color=CO LOR(RGB,0,.75,0),font=[HELVETICA,BOLD,10]):\nt6 := plots[textplot]([8. 45,3.3,`w'`],\n color=COLOR(RGB,.75,.1,.75),font=[HELVETICA, BOLD,10]):\nt7 := plots[textplot]([5.2,4.55,`(v + w)' = v' + w'`],\n \+ color=brown,font=[HELVETICA,BOLD,10]):\nt8 := plots[textplot] ([[14,7.3,`P`],[11.2,3.8,`O`]],\n color=black,font=[HELVETIC A,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,\n \+ t1,t2,t3,t4,t5,t6,t7,t8],axes=none);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }