{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 311 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 316 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 317 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 319 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 320 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 321 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 322 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 16 128 0 0 1 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 128 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Tit le" 0 18 1 {CSTYLE "" -1 -1 "" 1 20 0 0 0 0 0 1 2 0 0 2 0 0 0 }3 1 0 -1 5 3 0 0 0 0 0 0 18 0 }{PSTYLE "" 5 256 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 18 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 18 266 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 18 267 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 18 268 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 14 "Maple Basics " }}{PARA 18 "" 0 "" {TEXT -1 8 "Lesson 1" }}{PARA 18 "" 0 "" {TEXT -1 7 "Vectors" }} {PARA 266 "" 0 "" {TEXT -1 37 "*revised from a Maple worksheet from " }}{PARA 268 "" 0 "" {TEXT -1 25 "Department of Mathematics" }}{PARA 267 "" 0 "" {TEXT -1 31 "N.C. State University 1999-2000" }}{PARA 257 "" 0 "" {TEXT 302 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduc tion" }}{PARA 0 "" 0 "" {TEXT -1 130 "Vector calculations using Maple \+ V are simple. This lesson introduces the basic Maple V commands neede d to do vector calculations." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 " Discussion" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "Preliminaries" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Ve ctor calculations using Maple are easy. The " }{TEXT 257 15 "main dif ference" }{TEXT -1 52 " between your textbook and a Maple worksheet is the " }{TEXT 258 8 "notation" }{TEXT -1 72 " used to denote vectors. \+ Thus while your textbook will use the notation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 261 1 "A" }{TEXT -1 4 " = <" } {XPPEDIT 18 0 "1,2,3" "6%\"\"\"\"\"#\"\"$" }{TEXT -1 6 " >, " } {TEXT 262 1 "B" }{TEXT -1 4 " = <" }{XPPEDIT 18 0 "5,0,5" "6%\"\"&\"\" !\"\"&" }{TEXT -1 4 ">, " }{TEXT 263 1 "C" }{TEXT -1 4 " = <" } {XPPEDIT 256 0 "x,y,z" "6%%\"xG%\"yG%\"zG" }{TEXT -1 1 ">" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "for vectors, Maple uses the notation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 " " {TEXT -1 1 " " }{TEXT 0 36 "A:=[1,2,3]; B:=[5,0,5]; C:= [x,y,z];" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Maple avo ids using the " }{TEXT 265 17 "angular brackets " }{TEXT -1 60 "< and \+ > to denote a vector in order to avoid confusion with " }{TEXT 266 10 "less than " }{TEXT -1 3 "and" }{TEXT 299 16 " greater than. " } {TEXT -1 100 "As you will discover below, the Maple syntax follows the syntax you have learned in your textbook. " }{TEXT 264 1 " " }{TEXT -1 32 "Thus if you want to add vectors " }{TEXT 259 1 "A" }{TEXT -1 5 " and " }{TEXT 260 1 "B" }{TEXT -1 39 ", then in your worksheet you wi ll write" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " " }{TEXT 0 5 "A + B" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "just as you would write \+ on paper with a pencil. Throughout the worksheet when we want to tal k about a vector, we will use the notation " }{TEXT 298 9 " " }{TEXT -1 17 " for the vector " }{TEXT 300 10 "ai +bj +ck" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "There are a few commands for which you will need special notation , and these Maple V commands are part of the Maple V Linear Algebra p ackage called " }{TEXT 0 6 "linalg" }{TEXT -1 40 ". This package is ma de available by the " }{TEXT 0 12 "with(linalg)" }{TEXT -1 10 " comma nd." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Defining Vectors" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "To begin, enter the following vectors " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 261 "" 0 "" {TEXT 301 47 "A = <1, 2, 3>, B = <5, 0, 5>, \+ C = " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "in order to illustrate some of the basic Maple V commands that deal with vectors. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "vector" {MPLTEXT 1 0 13 "A := [1,2,3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "B := [5,0,5];" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The vector " } {TEXT 267 1 "C" }{TEXT -1 57 " has variable components and is entered in the same way:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "C := [x,y,z];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Maple as usual keeps trac k of these assignments that you have made. We can check this as follo ws:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "B;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "C;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "It is easy to assign va lues to the variables in a vector." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " subs(\{x=1,y=2,z=3\},C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Note this does not change the value of C as far as Maple is concerned though." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "C;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "Vector Addition" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 47 "To add two vec tors using Maple simply form the " }{TEXT 268 3 "sum" }{TEXT -1 20 " o f the two vectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "A+B;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "A+B-C;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "This technique will be useful in constructing a vector from one point to another." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 11 "PROBLEM : " }{TEXT -1 46 "Find the vector from P=(1,4,-6) to Q=(11,-5,0)" }{TEXT 288 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 269 10 "Solution: " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "This vector, call it vec(PQ), is constructed by subtr acting the coordinates of P from the coordinates of Q." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "vec( PQ):=[11,-5,0]-[1,4,-6];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Scalar multiplication" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "O ne can multiply a vector by a scalar (scalar multiplication) just as you would on paper. Of course we use the " }{TEXT 0 1 "*" }{TEXT -1 60 " symbol to denote multiplication. Here are a few examples." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(1/10)*A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "[1,2,3]*( 1/10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "3*C;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Pi*B;" }}}{EXCHG {PARA 256 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Length of a Vector" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Recall that if " }{TEXT 271 1 "V" }{TEXT -1 5 " = <" } {XPPEDIT 18 0 "a,b,c" "6%%\"aG%\"bG%\"cG" }{TEXT -1 22 "> then the le ngth of " }{TEXT 270 1 "V" }{TEXT -1 16 ", denoted by ||" }{TEXT 272 1 "V" }{TEXT -1 7 "|| , is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 50 " \+ ||" }{TEXT 273 1 "V" }{TEXT -1 5 "|| = " }{XPPEDIT 18 0 "sqrt(a^2+b ^2+c^2)" "6#-%%sqrtG6#,(*$%\"aG\"\"#\"\"\"*$%\"bG\"\"#F**$%\"cG\"\"#F* " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We can calulate this value using the Maple V command " } {TEXT 0 4 "norm" }{TEXT -1 148 ". The words \"norm of a vector\" are \+ often used in place of the phrase \"length of a vector\", and Maple V uses this terminology. There are in fact " }{TEXT 274 29 "many diff erent norm functions" }{TEXT -1 117 ", and Maple knows all about them. We want to always use here the Euclidean norm, and so the syntax for \+ the command is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{TEXT 0 14 "norm(vector,2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "where the 2 tells Maple t o use the Euclidean norm. Here are a few examples of norm calculation s:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "norm(A,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "norm(B,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "norm(C,2) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "norm(5*A,2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "norm(5*C,2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 7 "PROBLEM" }{TEXT -1 60 ": Find the length of the vector that points from the po int " }{TEXT 0 12 "R =(2,-1,16)" }{TEXT -1 14 " to the point " }{TEXT 0 10 "S=(7,0,-9)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The vector from R to S is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "vec(RS):=[7 ,0,-9]-[2,-1,16];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 34 "The length of this vector is then:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "length _vec(RS):=norm(vec(RS),2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Inner (Dot) and Cross P roducts" }}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "dot pr oduct " {TEXT -1 69 "The vector operations dot product and cross produ ct are available in " }{TEXT 0 6 "linalg" }{TEXT -1 82 ", with a sligh t change in notation. The commands have the easy names to remember: " }{TEXT 276 1 " " }{TEXT 0 9 "crossprod" }{TEXT 305 1 " " }{TEXT -1 21 "for the cross product" }{TEXT 304 2 ", " }{TEXT -1 4 "but " }{TEXT 0 9 "innerprod" }{TEXT 306 1 " " }{TEXT -1 30 " for the dot product. N otice" }{TEXT 303 1 " " }{TEXT -1 92 "the change in name for the dot p roduct. The reason is that Maple also has a command called " }{TEXT 0 7 "dotprod" }{TEXT -1 104 " , but it is used in complex vectors spac es and we want the real vector space dot product. This is the " } {TEXT 0 9 "innerprod" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Here are a few example calculat ions that illustrate the use of these commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 14 "First observe " }{TEXT 0 9 "innerprod" }{TEXT -1 11 " i n action." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "d otprod" {MPLTEXT 1 0 15 "innerprod(A,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "innerprod(B,A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "innerprod([5,0,5],[1,2,3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "innerprod(A,C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "cross product" {TEXT -1 28 "Now for some examples using " }{TEXT 0 9 "crossprod" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "crossprod" {MPLTEXT 1 0 15 "crossprod(A,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "cros sprod(B,A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "crossprod([1 ,2,3],[5,0,5]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Now try computing the cross product of " }{TEXT 0 1 "A" }{TEXT -1 6 " with " }{TEXT 0 1 "C" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "cros sprod(A,C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 277 8 "PROBLEM:" }{TEXT -1 75 " Find a vector that is orthogon al to the plane containing the three points " }{TEXT 0 9 "P=(1,2,3)" } {TEXT -1 2 ", " }{TEXT 0 9 "Q=(4,5,6)" }{TEXT -1 6 ", and " }{TEXT 0 11 "R=(-2,7,11)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Since the cross product of two vectors produce s a vector orthogonal to the plane spanned by the two vectors, the ans wer to our problem is " }{TEXT 0 26 "crossprod(vec(pq),vec(pr))" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "vec(pq):=[4,5,6]-[1,2,3];" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 27 "vec(pr):=[-2,7,11]-[1,2,3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "So now co mpute the cross product of these two vectors:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "orthogonal_ vector:=crossprod(vec(pq),vec(pr));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Note that any non-zero multiple of this vector will also be orthogonal to the plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Angle between Two Vectors" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "In your textbook you have learned that the dot product of vector " } {TEXT 278 1 "A" }{TEXT -1 12 " and vector " }{TEXT 279 1 "B" }{TEXT -1 20 " can be expressed as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " dotprod( " }{TEXT 280 3 "A,B" }{TEXT -1 4 ")=||" }{TEXT 281 1 "A" }{TEXT -1 5 " || ||" }{TEXT 282 1 "B" }{TEXT -1 7 "|| cos(" }{XPPEDIT 18 0 "theta" " 6#%&thetaG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 76 " is the angle between the two vectors. Solving this equa tion for the angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 20 " yields the formula " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " cos(" } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 12 ") = dotprod(" }{TEXT 283 3 "A,B" }{TEXT -1 5 ")/(||" }{TEXT 284 1 "A" }{TEXT -1 5 "|| ||" } {TEXT 285 1 "B" }{TEXT -1 3 "||)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "Since the angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 18 " must be between -" }{XPPEDIT 18 0 "Pi" "6# %#PiG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 68 ", we can use the inverse cosine function to solve this equation for \+ " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 26 ". Maple has the f unction " }{TEXT 0 5 "angle" }{TEXT 286 1 " " }{TEXT -1 53 "for doing \+ this computation. Here are some examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "angle([1,0,0],[0, 0,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "angle([1,0,0],[1, 0,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "angle(A,A);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 11 "Example \+ 1: " }{TEXT -1 52 "Find the unit vector in the direction of the vector " }{TEXT 0 12 "A = <-2,5,3>" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 307 9 "Solution:" }}{PARA 0 " " 0 "" {TEXT -1 156 " From our textbook we know the method of doing th is. We divide vector A by its length. That is, we multiply vector A \+ by the reciprocal of the length of A." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "A:=[1,2,3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "U := (1/norm(A,2))*A ;" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "As a check we can verify that vector U has leng th one:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "norm(U,2);" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 10 "E xample 2:" }{TEXT -1 5 " Let " }{TEXT 0 12 "F = <-1,1,3>" }{TEXT -1 5 " and " }{TEXT 0 11 "G = <2,4,3>" }{TEXT -1 29 " be vectors. Find ve ctors " }{TEXT 0 3 "Par" }{TEXT -1 6 " and " }{TEXT 0 4 "Perp" } {TEXT -1 14 " such that " }{TEXT 0 14 "F = Par + Perp" }{TEXT -1 18 ", and such that " }{TEXT 0 3 "Par" }{TEXT -1 16 " is parallel to " }{TEXT 0 1 "G" }{TEXT -1 5 " and " }{TEXT 0 4 "Perp" }{TEXT -1 21 " is perpendicular to " }{TEXT 0 1 "G" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 2 " " }}}{SECT 1 {PARA 262 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " The first step is to define the vectors " }{TEXT 308 1 "F" }{TEXT -1 5 " and " }{TEXT 309 1 "G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F := [-1,1,3]; G := \+ [2,4,3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The vector " }{XPPEDIT 18 0 "F[parallel]" "6#&%\"FG6#%)pa rallelG" }{TEXT -1 39 " is obtained by taking the component of" } {TEXT 310 2 " F" }{TEXT -1 21 " in the direction of " }{TEXT 311 1 "G " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Fparallel := (innerprod(F,G)/innerprod(G,G)) *G;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The vector " }{XPPEDIT 18 0 "F[perp]" "6#&%\"FG6#%%perpG" } {TEXT -1 28 " is the difference between " }{TEXT 312 1 "F" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "F[parallel]" "6#&%\"FG6#%)parallelG" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "Fperp := F-Fparallel;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "As a check." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "innerprod(G,Fperp);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "crossprod(G,Fparallel);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 10 "Example 3:" }{TEXT -1 72 " F ind the equation of the plane that passes through the three points " }{TEXT 0 21 "P=(1,0,1), Q=(1,-1,3)" }{TEXT -1 6 ", and " }{TEXT 0 10 " R=(3,0,-1)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}} {SECT 1 {PARA 263 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "To write down the equation of a plane we need: (i) a point " }{XPPEDIT 18 0 "P[0];" "6#&%\"PG6#\"\"!" }{TEXT 317 1 " " } {TEXT -1 39 "on the plane, and (ii) a normal vector " }{TEXT 316 2 " N " }{TEXT -1 51 " for the plane. We have three points to choose for " } {XPPEDIT 18 0 "P[0];" "6#&%\"PG6#\"\"!" }{TEXT -1 16 ", so let's take \+ " }{XPPEDIT 18 0 "P[0];" "6#&%\"PG6#\"\"!" }{TEXT -1 132 "=P=(1,0,1). \+ To get a normal vector we take the cross product of any two vectors p arallel to the plane. Let U=vec(PQ) and V=vec(PR)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "U := [1,-1 ,3]-[1,0,1] ; V := [3,0,-1]-[1,0,1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Then for the normal vector " } {TEXT 315 2 "N " }{TEXT -1 29 "we take the cross product of " }{TEXT 313 2 " U" }{TEXT -1 6 " and " }{TEXT 314 1 "V" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "N := crossprod(U,V);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "The equation of the plane is thus:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " 2*(x-1)+4*(y-0)+2*(z-1)=0;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 10 " Example 4:" }{TEXT -1 91 " Find the area, correct to 10 digits, of th e triangle formed by the points with vertices:" }}{PARA 0 "" 0 "" {TEXT 0 7 " A = (" }{XPPEDIT 18 0 "-122,317,615" "6%,$\"$A\"!\"\"\"$< $\"$:'" }{TEXT 0 1 ")" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT 0 7 " \+ B = (" }{XPPEDIT 18 0 "217,314,617" "6%\"$<#\"$9$\"$<'" }{TEXT 0 1 ") " }{TEXT -1 6 ", and" }}{PARA 0 "" 0 "" {TEXT 0 7 " C = (" } {XPPEDIT 18 0 "3117,-217,615" "6%\"% " 0 "" {MPLTEXT 1 0 33 "AB:=[217,314,617]-[-122,317 ,615];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "AC:=[3117,-217,61 5]-[-122,317,615];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "area of a triangle" {TEXT -1 56 "The area of the parallelogram d etermined by the vectors " }{TEXT 320 2 "AB" }{TEXT -1 5 " and " } {TEXT 321 2 "AC" }{TEXT -1 50 " is equal to the length of the crossp roduct of " }{TEXT 322 2 "AB" }{TEXT -1 6 " and " }{TEXT 323 2 "AC" }{TEXT -1 74 ". The area of the triangle is one-half of the area of \+ the parallelogram." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "AreaTriangle := norm(crossprod(AB,AC),2)/2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%,10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "" 1 "m a242\\example4.mws" "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "volume of a parallopiped" {TEXT 293 10 "Example 5:" } {TEXT -1 61 " Find the volume of the parallelopiped formed by the vec tors" }}{PARA 0 "" 0 "" {TEXT 0 14 " A = <1,2,3>" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT 0 15 " B = <-1,0,1>" }{TEXT -1 5 ", and" }} {PARA 0 "" 0 "" {TEXT 0 14 " C = <2,5,5>" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 1 {PARA 265 "" 0 "" {TEXT -1 9 "Soluti on:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Area := abs(innerprod([1,2,3 ],crossprod([-1,0,1],[2,5,5])));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Summary" }}{PARA 0 "" 0 "" {TEXT -1 58 "These commands are contained in the Maple libra ry package " }{TEXT 0 6 "linalg" }{TEXT -1 30 ". You must enter the c ommand " }{TEXT 0 13 "with(linalg):" }{TEXT -1 22 " in order to use th em." }}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 4 "norm" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The Maple " }{TEXT 0 4 "norm" }{TEXT -1 33 " command can be used to find the " }{TEXT 297 6 "length" }{TEXT -1 61 " of the vector. If we want to find the lengt h of the vector " }{TEXT 0 1 "A" }{TEXT -1 16 ", the command is" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 0 21 " norm(A,2);" }} {PARA 0 "" 0 "" {TEXT -1 11 "(Note: The " }{TEXT 0 1 "2" }{TEXT -1 40 " tells Maple to use the Euclidean norm.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "See also " }{HYPERLNK 17 "M aple's norm" 2 "linalg,norm" "" }{TEXT -1 6 " help." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "innerprod" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The Maple " }{TEXT 0 9 "innerpr od" }{TEXT -1 36 " command can be used to compute the " }{TEXT 296 11 "dot product" }{TEXT -1 72 " of the two vectors. If we want to find t he dot product of the vectors " }{TEXT 0 1 "A" }{TEXT -1 5 " and " } {TEXT 0 1 "B" }{TEXT -1 17 ", the command is " }}{PARA 0 "" 0 "" {TEXT 0 26 " innerprod(A,B);" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "See also " }{HYPERLNK 17 "Mapl e's innerprod" 2 "innerprod" "" }{TEXT -1 6 " help." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "crossprod" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The Maple " }{TEXT 0 9 "crosspr od" }{TEXT -1 36 " command can be used to compute the " }{TEXT 295 13 "cross product" }{TEXT -1 74 " of the two vectors. If we want to find the cross product of the vectors " }{TEXT 0 1 "A" }{TEXT -1 5 " and \+ " }{TEXT 0 1 "B" }{TEXT -1 17 ", the command is " }}{PARA 0 "" 0 "" {TEXT 0 27 " crossprod(A,B);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "See also " }{HYPERLNK 17 "Maple's crossprod" 2 "linalg,crossprod" " " }{TEXT -1 6 " help." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 5 "angle" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The Maple " }{TEXT 0 5 "angle" } {TEXT -1 33 " command can be used to find the " }{TEXT 294 5 "angle" } {TEXT -1 80 " between the two vectors. If we want to find the angle b etween the two vectors " }{TEXT 0 1 "A" }{TEXT -1 5 " and " }{TEXT 0 1 "B" }{TEXT -1 16 ", the command is" }}{PARA 0 "" 0 "" {TEXT 0 24 " \+ angle(A,B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "See also " }{HYPERLNK 17 "Maple's angle" 2 "linalg,angle" "" }{TEXT -1 6 " help." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "4 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 }