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}{TEXT 259 11 "dot 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#&%\"zG 6#\"\"\"" }{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 40 " i s the real number (or scalar) given by" }}{PARA 256 "" 0 "" {TEXT 271 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT 274 4 " . v" } {XPPEDIT 18 0 "``[2] = x[1]*x[2]+y[1]*y[2]+z[1]*z[2];" "6#/&%!G6#\"\"# ,(*&&%\"xG6#\"\"\"F-&F+6#F'F-F-*&&%\"yG6#F-F-&F26#F'F-F-*&&%\"zG6#F-F- &F86#F'F-F-" }{TEXT -1 15 " ------- (i). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 18 "__________________" }{TEXT -1 18 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 9 "Example I" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {TEXT 281 1 "u" }{TEXT -1 5 " = 4 " }{TEXT 275 1 "i" }{TEXT -1 4 " -3 \+ " }{TEXT 276 1 "j" }{TEXT -1 5 " + 2 " }{TEXT 277 1 "k" }{TEXT -1 5 " \+ and " }{TEXT 282 1 "v" }{TEXT -1 5 " = 5 " }{TEXT 278 1 "i" }{TEXT -1 5 " + 2 " }{TEXT 279 1 "j" }{TEXT -1 3 " - " }{TEXT 280 1 "k" }{TEXT -1 6 " then " }{TEXT 273 5 "u . v" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "` `(4)*``(5)+``(-3)*``(2)+``(2)*``(-1)" "6#,(*&-%!G6#\"\"%\"\"\"-F&6#\" \"&F)F)*&-F&6#,$\"\"$!\"\"F)-F&6#\"\"#F)F)*&-F&6#F5F)-F&6#,$F)F2F)F)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "20-6-2 = 12" "6#/,(\"#?\"\"\"\"\"'! \"\"\"\"#F(\"#7" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 353 254 254 {PLOTDATA 2 "63-%'CURVESG6'7$7$$\"\"!F)F(7$$\"\" #F)$\"+;;5kM!\"*7%7$$\"+)H()p'*)!#5$\"+gx7`;F/7$$\"\"\"F)$\"+330KbJF/FA-FF6&FHFIF(F(FL-F$6$7)7$$\"3I+++PSDg')!#=$\"3 3+++,+++]F\\q7$$\"3%)*****>W?:>)F\\q$\"3++++kVwNdF\\q7$$\"3k******HWWg wF\\q$\"3[+++)4wyU'F\\q7$$\"3=+++5y1rqF\\q$\"3U+++8y1rqF\\q7$$\"3C+++& 4wyU'F\\q$\"3!)*****>VW/m(F\\q7$$\"3%)*****>Okdt&F\\q$\"3-+++W/_\">)F \\q7$$\"3;+++-+++]F\\qFjp-FF6&FHF)F)F)-%%TEXTG6&7$$\"#&)!\"#F+Q\"v6\"F E-%%FONTG6%%*HELVETICAG%%BOLDG\"#5-F^s6&7$$\"#F!\"\"$\"#9FatFdsFEFfs-F ^s6&7$$\"#'*Fcs$\"#>FatQ\"1FesFE-Fgs6$Fis\"\")-F^s6&7$$\"$$GFcs$\"#8Fa tQ\"2FesFEF\\u-F^s6&7$$\"#NFatFjuQ)v~~~-~~vFesFdpFfs-F^s6&7$$\"$l$Fcs$ \"#MFatQ+2~~~~~~~~1FesFdpF\\u-F^s6&7$$FTFat$\"#cFcsQ\"qFesF[s-Fgs6$%'S YMBOLGF[t-F^s6&7$$FcsFatF(Q\"OFesF[s-Fgs6$FisF[t-F^s6&7$FitF`vQ\"AFesF [sFdw-F^s6&7$$\"#^Fat$\"#JFatQ\"BFesF[sFdw-%+AXESLABELSG6$Q!FesFex-%*A XESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGF]y" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "Applying the " }{TEXT 259 11 "cosine \+ rule" }{TEXT -1 28 " to the triangle 0AB gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "AB^2 = OA^2+OB^2-2*OA*OB*cos*theta;" "6 #/*$%#ABG\"\"#,(*$%#OAGF&\"\"\"*$%#OBGF&F**,F&F*F)F*F,F*%$cosGF*%&thet aGF*!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 4 " || " }{TEXT 284 1 "v" }{XPPEDIT 18 0 "` `[2]-``;" "6#,&&%!G6#\"\"#\"\"\"F%!\"\"" }{TEXT 285 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 6 " = || " }{TEXT 286 1 "v" }{XPPEDIT 18 0 "` `[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%! G\"\"#" }{TEXT -1 7 " + || " }{TEXT 287 1 "v" }{XPPEDIT 18 0 "``[2]; " "6#&%!G6#\"\"#" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\" #" }{TEXT -1 8 " - 2 || " }{TEXT 288 1 "v" }{XPPEDIT 18 0 "``[1]" "6#& %!G6#\"\"\"" }{TEXT -1 7 " || || " }{TEXT 289 1 "v" }{XPPEDIT 18 0 "`` [2]" "6#&%!G6#\"\"#" }{TEXT -1 4 " || " }{XPPEDIT 18 0 "cos*theta;" "6 #*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 6 " 2 || " }{TEXT 290 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 7 " || || " }{TEXT 291 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 4 " || " } {XPPEDIT 18 0 "cos*theta;" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 8 " \+ = || " }{TEXT 292 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 7 " + \+ || " }{TEXT 293 1 "v" }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{XPPEDIT 18 0 "``-`` " "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 4 " || " }{TEXT 294 1 "v" } {XPPEDIT 18 0 "``[2]-``;" "6#,&&%!G6#\"\"#\"\"\"F%!\"\"" }{TEXT 295 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " ||" } {XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{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[2]-x[1])^ 2+(y[2]-y[1])^2+(z[2]-z[1])^2)" "6#/%!G,(-F$6#,(*$&%\"xG6#\"\"\"\"\"#F -*$&%\"yG6#F-F.F-*$&%\"zG6#F-F.F-F--F$6#,(*$&F+6#F.F.F-*$&F16#F.F.F-*$ &F56#F.F.F-F--F$6#,(*$,&&F+6#F.F-&F+6#F-!\"\"F.F-*$,&&F16#F.F-&F16#F-F LF.F-*$,&&F56#F.F-&F56#F-FLF.F-FL" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 2 *x[1]*x[2] + 2*y[1]*y[2] + 2*z[1]*z[2]" "6#/%!G,(*(\"\"#\"\"\"&%\"xG6# F(F(&F*6#F'F(F(*(F'F(&%\"yG6#F(F(&F06#F'F(F(*(F'F(&%\"zG6#F(F(&F66#F'F (F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 57 "Using the formula (i) for the dot product of the vector s " }{TEXT 306 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 307 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" } {TEXT -1 10 ", we have " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 6 " 2 || " }{TEXT 304 1 "v" }{XPPEDIT 18 0 "``[1]" "6# &%!G6#\"\"\"" }{TEXT -1 7 " || || " }{TEXT 305 1 "v" }{XPPEDIT 18 0 "` `[2]" "6#&%!G6#\"\"#" }{TEXT -1 4 " || " }{XPPEDIT 18 0 "cos*theta;" " 6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 6 " = 2 " }{TEXT 301 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 1 " " }{TEXT 300 1 ". " }{TEXT -1 1 " " }{TEXT 302 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\" \"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "This shows that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 298 1 "v" }{XPPEDIT 18 0 "` `[1]" "6#&%!G6#\"\"\"" }{TEXT -1 1 " " }{TEXT 339 1 "." }{TEXT -1 1 " \+ " }{TEXT 299 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 7 " = || " }{TEXT 296 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 7 " || || " }{TEXT 297 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6# \"\"#" }{TEXT -1 4 " || " }{XPPEDIT 18 0 "cos*theta;" "6#*&%$cosG\"\" \"%&thetaGF%" }{TEXT -1 15 " ------- (ii). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 303 13 "_____________" }{TEXT -1 16 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "T hus " }{TEXT 259 120 "the dot product of two vectors is the product of the lengths of the two vectors and the cosine of the angle between th em" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 15 ": If the angle " }{XPPEDIT 18 0 "thet a" "6#%&thetaG" }{TEXT -1 25 " between the two vectors " }{TEXT 340 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " } {TEXT 341 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 20 " \+ is 0, that is, if " }{TEXT 342 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6# \"\"\"" }{TEXT -1 5 " and " }{TEXT 343 1 "v" }{XPPEDIT 18 0 "``[2]" "6 #&%!G6#\"\"#" }{TEXT -1 20 " are parallel, then " }{XPPEDIT 18 0 "cos* theta = cos*0;" "6#/*&%$cosG\"\"\"%&thetaGF&*&F%F&\"\"!F&" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 9 ", so that" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 344 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\" \"" }{TEXT -1 1 " " }{TEXT 346 1 "." }{TEXT -1 1 " " }{TEXT 345 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 7 " = || " }{TEXT 347 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 7 " || || \+ " }{TEXT 348 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 4 " ||." }}{PARA 0 "" 0 "" {TEXT -1 36 "In particular, for a single vect or " }{TEXT 357 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 " " } {TEXT 352 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\" \"\"" }{TEXT -1 1 " " }{TEXT 355 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 356 1 "k" }{TEXT -1 3 ", " }}{PARA 256 "" 0 "" {TEXT 349 2 " v" }{XPPEDIT 18 0 "``[1] " "6#&%!G6#\"\"\"" }{TEXT 354 4 " . v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6 #\"\"\"" }{TEXT -1 6 " = || " }{TEXT 350 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"# " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 5 " || " }{TEXT 351 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6 #\"\"\"" }{TEXT -1 6 " || = " }{XPPEDIT 18 0 "sqrt(``)" "6#-%%sqrtG6#% !G" }{TEXT -1 1 "(" }{TEXT 353 2 " v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6# \"\"\"" }{TEXT 359 4 " . v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 5 " ) = " }{XPPEDIT 18 0 "sqrt(x[1]^2+y[1]^2+z[1]^2);" "6#-%% sqrtG6#,(*$&%\"xG6#\"\"\"\"\"#F+*$&%\"yG6#F+F,F+*$&%\"zG6#F+F,F+" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Equation (ii) can be rearranged in the form: " }}{PARA 256 "" 0 "" {TEXT 308 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 1 " " }{TEXT 337 1 "." }{TEXT -1 1 " " }{TEXT 309 1 "v" } {XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "cos*theta;" "6#*&%$cosG\"\"\" %&thetaGF%" }{TEXT -1 43 " = ___________ \+ " }}{PARA 256 "" 0 "" {TEXT -1 4 " || " }{TEXT 310 1 "v" }{XPPEDIT 18 0 "``[1];" "6#&%!G6#\"\"\"" }{TEXT -1 7 " || || " }{TEXT 311 1 "v" } {XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 4 " || " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Hence the angle " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 21 " between 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 5 " is " }}{PARA 256 "" 0 "" {TEXT 318 2 " v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 1 " " }{TEXT 338 1 "." }{TEXT -1 1 " " }{TEXT 319 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 18 0 "the ta = arccos;" "6#/%&thetaG%'arccosG" }{TEXT 323 2 " (" }{TEXT -1 15 " \+ ___________ " }{TEXT 324 1 ")" }{TEXT -1 22 " ------- (iii). \+ " }}{PARA 256 "" 0 "" {TEXT -1 4 " || " }{TEXT 320 1 "v" }{XPPEDIT 18 0 "``[1];" "6#&%!G6#\"\"\"" }{TEXT -1 7 " || || " }{TEXT 321 1 "v" } {XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 4 " || " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 322 11 "___________" }{TEXT -1 13 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 10 "Example II" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 10 "The angle " }{XPPEDIT 18 0 "theta" "6#%&t hetaG" }{TEXT -1 25 " between the two vectors " }{TEXT 325 1 "u" } {TEXT -1 5 " = 2 " }{TEXT 326 1 "i" }{TEXT -1 5 " - 2 " }{TEXT 327 1 " j" }{TEXT -1 3 " + " }{TEXT 328 1 "k" }{TEXT -1 5 " and " }{TEXT 329 1 "v" }{TEXT -1 5 " = 6 " }{TEXT 330 1 "i" }{TEXT -1 5 " + 3 " }{TEXT 331 1 "j" }{TEXT -1 5 " + 2 " }{TEXT 332 1 "k" }{TEXT -1 14 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta = arcc os;" "6#/%&thetaG%'arccosG" }{TEXT -1 2 "( " }{TEXT 336 5 "u . v" } {TEXT -1 7 " / || " }{TEXT 313 2 "u " }{TEXT -1 6 "|| || " }{TEXT 314 1 "v" }{TEXT -1 6 " || )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Since " }{TEXT 317 5 "u . v" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``(2)*``(6)+``(-2)*``(3) +``(1)*``(2) = 8" "6#/, (*&-%!G6#\"\"#\"\"\"-F'6#\"\"'F*F**&-F'6#,$F)!\"\"F*-F'6#\"\"$F*F**&-F '6#F*F*-F'6#F)F*F*\"\")" }{TEXT -1 7 ", || " }{TEXT 315 2 "u " } {TEXT -1 5 "|| = " }{XPPEDIT 18 0 "sqrt(4+4+1)=3" "6#/-%%sqrtG6#,(\"\" %\"\"\"F(F)F)F)\"\"$" }{TEXT -1 9 " and || " }{TEXT 316 2 "v " } {TEXT -1 5 "|| = " }{XPPEDIT 18 0 "sqrt(36+9+4)=7" "6#/-%%sqrtG6#,(\"# O\"\"\"\"\"*F)\"\"%F)\"\"(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "theta = arccos(8/(``(3)*``(7))" "6#/%&thetaG -%'arccosG6#*&\"\")\"\"\"*&-%!G6#\"\"$F*-F-6#\"\"(F*!\"\"" }{XPPEDIT 18 0 "`` = arccos(8/21)" "6#/%!G-%'arccosG6#*&\"\")\"\"\"\"#@!\"\"" } {TEXT -1 1 " " }{TEXT 335 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "67.6^o " "6#)-%&FloatG6$\"$w'!\"\"%\"oG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The dot product of two ve ctors can be calculated using the procedure " }{TEXT 0 7 "dotprod" } {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 0 139 "with(linalg):\nu := vector([2,-2,1]);\nv := vector([6,3,2]);\nthe ta := arccos(dotprod(u,v)/(norm(u,2)*norm(v,2)));\nevalf(%);\nevalf(%* 180/Pi); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'vectorG6#7%\"\"# !\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7%\"\" '\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG-%'arccosG6## \"\")\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'>q*z6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+<7tgn!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Alternatively, the procedure " } {TEXT 0 10 "DotProduct" }{TEXT -1 8 " in the " }{TEXT 0 13 "LinearAlge bra" }{TEXT -1 22 " package can be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "u := < 2 | -2 | 1 >; \nv := < 6 | 3 | 2 >;\ntheta := arccos(LinearAlgebra[DotProduct](u,v)/ \n (LinearAlgebra[Norm](u,2)*LinearAlgebra[Norm](v,2)));\nevalf(%) ;\nevalf(%*180/Pi); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'RTABL EG6$\"*K@%[9-%'VECTORG6#7%\"\"#!\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'RTABLEG6$\"*OG,X\"-%'VECTORG6#7%\"\"'\"\"$\"\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG-%'arccosG6##\"\")\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'>q*z6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+<7tgn!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 19 "Orthogonal vectors " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 12 "Two vectors " } {TEXT 364 1 "u" }{TEXT -1 5 " and " }{TEXT 365 1 "v" }{TEXT -1 5 " are " }{TEXT 259 10 "orthogonal" }{TEXT -1 15 " provided that " }{TEXT 363 5 "u . v" }{TEXT -1 6 " = 0. " }}{PARA 0 "" 0 "" {TEXT -1 21 "Supp ose that neither " }{TEXT 358 1 "u" }{TEXT -1 5 " nor " }{TEXT 360 1 " v" }{TEXT -1 32 " is the zero vector, so that || " }{TEXT 361 1 "u" } {TEXT -1 3 " ||" }{XPPEDIT 18 0 "``<>0" "6#0%!G\"\"!" }{TEXT -1 8 " an d || " }{TEXT 362 1 "v" }{TEXT -1 3 " ||" }{XPPEDIT 18 0 "``<>0" "6#0% !G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {TEXT 366 5 "u . v" }{TEXT -1 6 " = || " }{TEXT 367 1 "u" }{TEXT -1 7 " || || " }{TEXT 368 1 "v" }{TEXT -1 4 " || " }{XPPEDIT 18 0 "cos*thet a;" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 22 " is the angle between " }{TEXT 369 1 "u" }{TEXT -1 5 " and " }{TEXT 370 1 "v" }{TEXT -1 18 ", it foll ows that " }{XPPEDIT 18 0 "cos*theta = 0;" "6#/*&%$cosG\"\"\"%&thetaGF &\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " } {XPPEDIT 18 0 "theta=Pi/2" "6#/%&thetaG*&%#PiG\"\"\"\"\"#!\"\"" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "90^o" "6#)\"#!*%\"oG" }{TEXT -1 7 ", thus " }{TEXT 371 1 "u" }{TEXT -1 5 " and " }{TEXT 372 1 "v" }{TEXT -1 5 " are " }{TEXT 259 13 "perpendicular" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 139 "(The zero vector is orthogonal to every vector, \+ but it does not make sense to say that the zero vector is perpendicula r to another vector.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 383 7 "Example" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 373 1 "u" }{TEXT -1 3 " = " }{TEXT 374 1 "i" }{TEXT -1 3 " + " }{TEXT 375 1 "j" }{TEXT -1 3 " + " }{TEXT 376 1 "k" }{TEXT -1 5 " and " }{TEXT 377 1 "v" }{TEXT -1 3 " = " }{TEXT 378 1 "j" }{TEXT -1 3 " - " }{TEXT 379 1 "i" }{TEXT -1 7 ", then " }{TEXT 380 5 "u . v " }{TEXT -1 9 " = 0, so " }{TEXT 381 1 "u" }{TEXT -1 5 " and " }{TEXT 382 1 "v" }{TEXT -1 47 " are orthogonal (and therefore perpendicular). 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G\"\"\"*&F%F%\"\"$!\"\"F(" }{TEXT -1 1 " " }{TEXT 451 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 452 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 11 "The vector " }{TEXT 470 1 "u" }{TEXT -1 5 " = 4 " }{TEXT 471 1 "i" }{XPPEDIT 18 0 "``- 5" "6#,&%!G\"\"\"\"\"&! \"\"" }{TEXT -1 1 " " }{TEXT 472 1 "j" }{TEXT -1 5 " + 3 " }{TEXT 473 1 "k" }{TEXT -1 29 " can now be expressed as the " }{TEXT 259 3 "sum" }{TEXT -1 15 " of the vector " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {TEXT 464 1 "p" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-2/3" "6#,$*&\"\"#\" \"\"\"\"$!\"\"F(" }{TEXT -1 1 " " }{TEXT 454 1 "i" }{XPPEDIT 18 0 "``- 1/3" "6#,&%!G\"\"\"*&F%F%\"\"$!\"\"F(" }{TEXT -1 1 " " }{TEXT 455 1 "j " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT -1 1 " " }{TEXT 456 1 "k" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 14 "and the vector" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {TEXT 463 1 "q" }{TEXT -1 3 " = " }{TEXT 465 1 "u" }{XPPEDIT 18 0 "``- ``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 466 1 "p" }{TEXT -1 7 " = ( 4 " } {TEXT 460 1 "i" }{XPPEDIT 18 0 "``- 5" "6#,&%!G\"\"\"\"\"&!\"\"" } {TEXT -1 1 " " }{TEXT 461 1 "j" }{TEXT -1 5 " + 3 " }{TEXT 462 1 "k" } {TEXT -1 2 " )" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 2 "( " }{XPPEDIT 18 0 "-2/3" "6#,$*&\"\"#\"\"\"\"\"$!\"\"F(" } {TEXT -1 1 " " }{TEXT 457 1 "i" }{XPPEDIT 18 0 "``-1/3" "6#,&%!G\"\"\" *&F%F%\"\"$!\"\"F(" }{TEXT -1 1 " " }{TEXT 458 1 "j" }{TEXT -1 3 " + \+ " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {TEXT 459 1 "k" }{TEXT -1 6 " ) = " }{XPPEDIT 18 0 "14/3" "6#*&\"#9\" \"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 467 1 "i" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-14/3;" "6#,$*&\"#9\"\"\"\"\"$!\"\"F(" }{TEXT -1 1 " " }{TEXT 468 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "7/3" "6#*&\"\"(\"\" \"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 469 1 "k" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 9 "which is " }{TEXT 259 10 "orthogonal" } {TEXT -1 4 " to " }{TEXT 474 1 "p" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "As a check note that " }{TEXT 475 1 "p" }{TEXT -1 1 " " } {TEXT 477 1 "." }{TEXT -1 1 " " }{TEXT 476 1 "q" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "-28/9+14/9+14/9 = 0;" "6#/,(*&\"#G\"\"\"\"\"*!\"\"F)*& \"#9F'F(F)F'*&F+F'F(F)F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Note that, since " }{TEXT 478 1 "p" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "-1/3" "6#,$*&\"\"\"F%\"\"$!\"\"F'" }{TEXT -1 1 " " } {TEXT 479 1 "v" }{TEXT -1 39 ", its direction is opposite to that of \+ " }{TEXT 480 1 "v" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {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 15 "Find the angle " }{XPPEDIT 18 0 "t heta" "6#%&thetaG" }{TEXT -1 25 " between the two vectors " }{TEXT 481 1 "u" }{XPPEDIT 18 0 "`` = 5;" "6#/%!G\"\"&" }{TEXT -1 1 " " } {TEXT 483 1 "j" }{XPPEDIT 18 0 "``- 3" "6#,&%!G\"\"\"\"\"$!\"\"" } {TEXT 485 2 " k" }{TEXT -1 5 " and " }{TEXT 482 13 "v = i + j + k" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Ans. " }{XPPEDIT 18 0 " arccos(15/19)" "6#-%'arccosG6#*&\"#:\"\"\"\"#>!\"\"" }{TEXT -1 1 " " } {TEXT 484 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "37.9^o" "6#)-%&FloatG6 $\"$z$!\"\"%\"oG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "____ _______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________ _______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 15 "Find the angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 25 " between the two vectors " }{TEXT 486 1 "u" }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\"\"#" }{TEXT -1 1 " " }{TEXT 493 1 "i" }{XPPEDIT 18 0 "``+2;" "6#,&%!G\"\"\"\"\"#F%" }{TEXT -1 1 " " }{TEXT 487 1 "j" } {XPPEDIT 18 0 "``+``;" "6#,&%!G\"\"\"F$F%" }{TEXT 488 1 "k" }{TEXT -1 5 " and " }{TEXT 489 1 "v" }{XPPEDIT 18 0 "`` = 2" "6#/%!G\"\"#" } {TEXT -1 1 " " }{TEXT 490 1 "i" }{XPPEDIT 18 0 " ``+ 10" "6#,&%!G\"\" \"\"#5F%" }{TEXT -1 1 " " }{TEXT 491 1 "j" }{XPPEDIT 18 0 "``" "6#%!G " }{XPPEDIT 18 0 " -11" "6#,$\"#6!\"\"" }{TEXT -1 1 " " }{TEXT 492 1 " k" }{TEXT -1 3 ". " }}{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 -1 145 "Find the an gle between the diagonal of a cube and a diagonal of one of its faces \+ which has an end point in common with the diagonal of the cube. " }} {PARA 0 "" 0 "" {TEXT -1 39 "Hint: Use a cube which has the vectors " }{TEXT 494 1 "i" }{TEXT -1 2 ", " }{TEXT 495 1 "j" }{TEXT -1 6 ", and \+ " }{TEXT 496 1 "k" }{TEXT -1 26 " as three of its edges. " }}{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 30 "Find the vector projection of " } {TEXT 501 1 "u" }{TEXT -1 3 " = " }{TEXT 502 1 "i" }{XPPEDIT 18 0 "``+ 3;" "6#,&%!G\"\"\"\"\"$F%" }{TEXT -1 1 " " }{TEXT 503 1 "j" }{XPPEDIT 18 0 " ``+ 4" "6#,&%!G\"\"\"\"\"%F%" }{TEXT -1 1 " " }{TEXT 504 1 "k" }{TEXT -1 21 " in the direction of " }{TEXT 497 1 "v" }{XPPEDIT 18 0 " ``" "6#%!G" }{XPPEDIT 18 0 "`` = 10;" "6#/%!G\"#5" }{TEXT 498 2 " i" } {XPPEDIT 18 0 "``" "6#%!G" }{XPPEDIT 18 0 "``-11;" "6#,&%!G\"\"\"\"#6! \"\"" }{TEXT -1 1 " " }{TEXT 499 1 "j" }{XPPEDIT 18 0 "``- 2" "6#,&%!G \"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 500 1 "k" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Ans. " }{XPPEDIT 18 0 "7/45" "6#*&\"\"(\" \"\"\"#X!\"\"" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "10;" "6#\"#5" }{TEXT 505 2 " i" }{XPPEDIT 18 0 "``" "6#%!G" }{XPPEDIT 18 0 "``-11;" "6#,&%! G\"\"\"\"#6!\"\"" }{TEXT -1 1 " " }{TEXT 506 1 "j" }{XPPEDIT 18 0 "``- 2" "6#,&%!G\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 507 1 "k" }{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 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 45 "(a) Find the vector \+ projection of the vector " }{TEXT 508 1 "u" }{XPPEDIT 18 0 "`` = 2" "6 #/%!G\"\"#" }{TEXT -1 1 " " }{TEXT 509 1 "i" }{XPPEDIT 18 0 "`` - 3" " 6#,&%!G\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 510 1 "j" }{XPPEDIT 18 0 "`` -`` " "6#,&%!G\"\"\"F$!\"\"" }{TEXT 511 1 "k" }{TEXT -1 32 " \+ in the direction of the vector " }{TEXT 512 1 "v" }{XPPEDIT 18 0 "``" "6#%!G" }{XPPEDIT 18 0 "`` = ``;" "6#/%!GF$" }{TEXT 513 1 "i" } {XPPEDIT 18 0 " ``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " " }{TEXT 514 1 "j" }{XPPEDIT 18 0 "`` + ``" "6#,&%!G\"\"\"F$F%" }{TEXT 522 1 "k " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) For the vectors \+ " }{TEXT 515 1 "u" }{TEXT -1 5 " and " }{TEXT 516 1 "v" }{TEXT -1 33 " in part (a), express the vector " }{TEXT 521 1 "u" }{TEXT -1 22 " as \+ a sum of a vector " }{TEXT 517 1 "p" }{TEXT -1 24 " parallel to the ve ctor " }{TEXT 520 1 "v" }{TEXT -1 14 " and a vector " }{TEXT 518 1 "q " }{TEXT -1 18 " perpendicular to " }{TEXT 519 1 "v" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Ans. " }{TEXT 533 1 "p" }{TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "4/3;" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {TEXT 523 1 "v" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "4/3;" "6#*&\"\"%\"\" \"\"\"$!\"\"" }{TEXT -1 3 " ( " }{TEXT 527 1 "i" }{XPPEDIT 18 0 " ``-` `" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " " }{TEXT 528 1 "j" }{XPPEDIT 18 0 "`` + ``" "6#,&%!G\"\"\"F$F%" }{TEXT 529 2 "k " }{TEXT -1 1 ")" } {XPPEDIT 18 0 "`` = 4/3;" "6#/%!G*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 524 1 "i" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-4/3;" "6#,$*&\" \"%\"\"\"\"\"$!\"\"F(" }{TEXT -1 1 " " }{TEXT 525 1 "j" }{XPPEDIT 18 0 "``+4/3;" "6#,&%!G\"\"\"*&\"\"%F%\"\"$!\"\"F%" }{TEXT -1 1 " " } {TEXT 526 1 "k" }{TEXT -1 6 ". (b) " }{TEXT 534 1 "q" }{XPPEDIT 18 0 " `` = 2/3;" "6#/%!G*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 530 1 "i" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-5/3;" "6#,$*&\"\"&\"\"\"\"\"$! \"\"F(" }{TEXT -1 1 " " }{TEXT 532 1 "j" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-7/3;" "6#,$*&\"\"(\"\"\"\"\"$!\"\"F(" }{TEXT -1 1 " " }{TEXT 531 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 4 "" 0 "" {TEXT 260 13 "plot2 dvector " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 993 "plot2dvector := proc(point::\{Vector, vector, list\} , vect::\{Vector, vector, list\},head_width,head_relative_height)\n \+ local a, b, i, x, y, Cos, Sin, v, locopts, L;\n x := point[1];\n y := point[2];\n a := vect[1];\n b := vect[2];\n if type(vect,'li st') then \n return procname(point,Vector(2,[a-x,b-y]),\n \+ head_width,head_relative_height,args[5..-1])\n end if;\n L := eval f(sqrt(a^2+b^2));\n if a=0 and b=0 then return CURVES([]) end if;\n \+ Cos := evalf(a/L);\n Sin := evalf(b/L);\n v := [[[x,y],[x+a,y+b] ],[[x-1/2*head_width*Sin \n -head_relative_height*Cos*L+1/2*a,\n \+ y+1/2*head_width*Cos-head_relative_height*Sin*L+1/2*b],\n \+ [x+1/2*a,y+1/2*b], \n [x+1/2*head_width*Sin-head_relative_heig ht*Cos*L+1/2*a, \n y-1/2*head_width*Cos -head_relative_height*S in*L+1/2*b]]];\n v := evalf(v);\n locopts := [args[5 .. -1]];\n \+ locopts := \n convert(['style'='patchnogrid',op(locopts)],'PL OToptions');\n PLOT(CURVES(op(v),op(locopts)))\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 33 "dot product and the cos ine rule " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1096 "x1 := 2: y1 := evalf(2*tan(Pi/3)): \nx2 := 5: y2 \+ := evalf(5*tan(Pi/6)):\nd := evalf(Pi/36):\np1 := plot2dvector([0,0],[ x1,y1],.1,.03,\n color=blue,thickness=2): \np2 := plot2dvector([0,0],[x2,y2],.1,.03,\n \+ color=blue,thickness=2):\np3 := plot2dvector([x1,y1],[x2,y2],.1,.0 5,\n color=red,thickness=2):\np4 := plot([seq([cos(i *d),sin(i*d)],i=6..12)],color=black):\nt1 := plots[textplot]([[.85,2,` v`],[2.7,1.4,`v`]],\n color=blue,font=[HELVETICA,BO LD,10]):\nt2 := plots[textplot]([[.96,1.9,`1`],[2.83,1.3,`2`]],\n \+ color=blue,font=[HELVETICA,8]):\nt3 := plots[textplot]( [3.5,3.5,`v - v`],color=red,\n font=[HELVETICA,B OLD,10]):\nt4 := plots[textplot]([3.65,3.4,`2 1`],color=red,\n \+ font=[HELVETICA,8]):\nt5 := plots[textplot]([.5,.56 ,`q`],color=black,\n font=[SYMBOL,10]):\nt6 := plot s[textplot]([[-.2,0,`O`],[1.9,3.65,`A`],\n [5.1,3.1,`B`]],colo r=black,font=[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,t1,t2,t3,t4 ,t5,t6],axes=none);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "orthogonal vectors " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1064 "u := 'u': v := 'v': \np1 : = plottools[arrow]([0,0,0],[1,1,1],.02,.1,.05,\n cylindri cal_arrow,color=green):\np2 := plottools[arrow]([1,0,0],[0,1,0],.02,.1 ,.05,\n cylindrical_arrow,color=blue):\np3 := plots[point plot3d]([[0,0,0],[1,0,0],[1,1,0],[0,1,0],\n [0,0,0],[0,0,1],[1,0,1] ,[1,1,1],[0,1,1],[0,0,1]],\n style=line,color=red):\np4 := p lots[pointplot3d]([[1,0,0],[1,0,1]],\n style=line,color=red) :\np5 := plots[pointplot3d]([[0,1,0],[0,1,1]],\n style=line, color=red):\np6 := plots[pointplot3d]([[1,1,0],[1,1,1]],\n s tyle=line,color=red):\nt1 := plots[textplot3d]([.5,.5,.65,`u`],color=g reen,\n font=[HELVETICA,BOLD,10]):\nt2 := plots [textplot3d]([.6,.6,0,`v`],color=blue,\n font=[ HELVETICA,BOLD,10]):\nt3 := plots[textplot3d]([[1.2,-.1,-.1,`x`],\n [- .1,1.2,-.1,`y`],[-.1,-.1,1.2,`z`]],color=black,\n \+ font=[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,t1,t2,t3], \n axes=normal,\n tickmarks=[2,2,2],view=[-.1..1.2,-.1..1.2, -.1..1.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "scalar \+ and vector projection " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1188 "d := evalf(Pi/36):\nrt := evalf(s qrt(3)): \np1 := plot2dvector([0,0],[1,rt],.05,.04,\n \+ color=COLOR(RGB,0,.8,0),thickness=2):\np2 := plot([[0,-.021],[1,- .021]],style=line,color=red,thickness=2):\np3 := plottools[arrow]([0,0 ],[2,0],0,.05,.04,\n arrow,color=blue,thickness= 2):\np4 := plot([[1,0],[1,rt]],linestyle=2,color=navy):\np5 := plot([[ 1,.1],[1.07,.1],[1.07,0]],color=COLOR(RGB,.3,.3,.3)):\np6 := plot([seq ([0.25*cos(i*d),0.25*sin(i*d)],i=0..12)],color=black):\nt1 := plots[te xtplot]([.45,.96,`u`],color=COLOR(RGB,0,.8,0),\n \+ font=[HELVETICA,BOLD,10]):\nt2 := plots[textplot]([1.4,-.1,`v`],colo r=blue,\n font=[HELVETICA,BOLD,10]):\nt3 := plo ts[textplot]([.465,-.1,`u`],color=red,\n font=[ HELVETICA,BOLD,10]):\nt4 := plots[textplot]([.55,-.1,` \+ q`],color=red,\n font=[SYMBOL,10]):\nt5 := pl ots[textplot]([.45,-.1,` || || cos `],color=red,\n \+ font=[HELVETICA,10]):\nt6 := plots[textplot]([.14,.09, `q`],color=black,\n font=[SYMBOL,10]):\nplots[d isplay]([p1,p2,p3,p4,p5,p6,t1,t2,t3,t4,t5,t6],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1219 "d := evalf(Pi/36):\nrt := evalf(sqrt(3) ): \np1 := plot2dvector([0,0],[-1,rt],.05,.03,\n \+ color=COLOR(RGB,0,.8,0),thickness=2):\np2 := plot([[0,-.021],[-1,-.021 ]],style=line,color=red,thickness=2):\np3 := plot2dvector([0,0],[2,0], .05,.04,\n color=blue,thickness=2):\np4 := \+ plot([[-1,0],[-1,rt]],linestyle=2,color=navy):\np5 := plot([[-1,.1],[- .93,.1],[-.93,0]],color=COLOR(RGB,.3,.3,.3)):\np6 := plot([[0,0],[-1.2 ,0]],color=black):\np7 := plot([seq([0.22*cos(i*d),0.22*sin(i*d)],i=0. .24)],color=black):\nt1 := plots[textplot]([-.44,.96,`u`],color=COLOR( RGB,0,.8,0),\n font=[HELVETICA,BOLD,10]):\nt2 : = plots[textplot]([1,-.08,`v`],color=blue,\n fo nt=[HELVETICA,BOLD,10]):\nt3 := plots[textplot]([-.69,-.1,`u`],color=r ed,\n font=[HELVETICA,BOLD,10]):\nt4 := plots[t extplot]([-.26,-.1,`q`],color=red,\n font=[SYMB OL,10]):\nt5 := plots[textplot]([-.69,-.1,` || || |cos | `],color=red,\n font=[HELVETICA,10]):\nt6 := pl ots[textplot]([.06,.09,`q`],color=black,\n font =[SYMBOL,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3,t4,t5,t6 ],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1124 "d := evalf(Pi/36):\nrt := evalf(sqrt(3)): \np1 := plot2dvector([0,0],[1,rt],.05,.03,\n color=COL OR(RGB,0,.8,0),thickness=2):\np2 := plot2dvector([0,-.023],[1,-.023],. 06,.05,\n color=red,thickness=2):\np3 := p lottools[arrow]([0,0],[2,0],0,.05,.03,\n arrow ,color=blue,thickness=2):\np4 := plot([[1,0],[1,rt]],linestyle=2,color =navy):\np5 := plot([[1,.1],[1.07,.1],[1.07,0]],color=COLOR(RGB,.3,.3, .3)):\np6 := plot([seq([0.2*cos(i*d),0.2*sin(i*d)],i=0..12)],color=bla ck):\nt1 := plots[textplot]([.44,.96,`u`],color=COLOR(RGB,0,.8,0),\n \+ font=[HELVETICA,BOLD,10]):\nt2 := plots[textplot ]([1.4,-.1,`v`],color=blue,\n font=[HELVETICA,B OLD,10]):\nt3 := plots[textplot]([.5,-.13,`u . v v . v v`],color= red,\n font=[HELVETICA,BOLD,10]):\nt4 := plots[ textplot]([.44,-.13,`( )/( )`],color=red,\n \+ font=[HELVETICA,10]):\nt5 := plots[textplot]([.12,.09,`q`], color=black,\n font=[SYMBOL,10]):\nplots[displa y]([p1,p2,p3,p4,p5,p6,t1,t2,t3,t4,t5],axes=none);" }}}{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 }{RTABLE_HANDLES 144842132 145012836 }{RTABLE M7R0 I6RTABLE_SAVE/144842132X*%)anythingG6"6"[gl!$%!!!"$"$""#!"#"""6" } {RTABLE M7R0 I6RTABLE_SAVE/145012836X*%)anythingG6"6"[gl!$%!!!"$"$""'""$""#6" }