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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 0 "" }}{PARA 258 "" 0 "" {TEXT -1 
0 "" }{TEXT 257 0 "" }{TEXT 258 19 "MAPLE for Math 2210" }}{PARA 257 "
" 0 "" {TEXT -1 20 "Friday Sept 10, 2004" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 665 "     You may use any technology y
ou want to complete or check your homework solutions in our section, M
ath 2210-1.  Sometimes the point of the problem is to let you practice
 by hand, in which case you should do the computation by hand, but fee
l free to check it with technology.  Other times, the point of the pro
blem is the mathematical concept and you can let your graphing calcula
tor or the computer do the grungy details.  An especially good example
 of this is the homework problem from last week in which you were to f
ind the area of a messy quadrilateral - if you look at the posted solu
tions, I set up the problem and then had the computer do the messy mat
h." }}{PARA 0 "" 0 "" {TEXT -1 402 "     This handout is being made us
ing a package called MAPLE, which exists on our Math system (to which \+
you all have accounts), as well as at Marriott and Engineering. (You c
an also buy a version from the University bookstore for about $120.)  \+
I will be happy to help any students interested in using this software
 -which as you can see allows you to mix word processing with mathemat
ical computations." }}{PARA 0 "" 0 "" {TEXT -1 449 "     Today we'll u
se MAPLE to do work and illustrate concepts related to sections 11.4-1
1.6 of our text.  A typical sort of problem would be like %11.4 #15 fr
om last week, in which you were to find whether two lines intersected,
 and if so where.  If you understand how to set this problem up, it is
 easy to have the computer find the answer.  Here are four ways to att
ack the same problem, illustrating some of the concepts we've been tal
king about:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
260 8 "Problem:" }{TEXT 265 1 " " }{TEXT 261 44 " Let two lines be giv
en in symmetric form by" }}{PARA 259 "" 0 "" {TEXT -1 5 "     " }
{XPPEDIT 18 0 "L[1];" "6#&%\"LG6#\"\"\"" }{TEXT -1 3 ":  " }{XPPEDIT 
18 0 "x-2;" "6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 4 "  = " }{XPPEDIT 
18 0 "1/2*y+1/2;" "6#,&*(\"\"\"F%\"\"#!\"\"%\"yGF%F%*&F%F%F&F'F%" }
{TEXT -1 4 "  = " }{XPPEDIT 18 0 "1/3*z-1;" "6#,&*(\"\"\"F%\"\"$!\"\"%
\"zGF%F%F%F'" }{TEXT -1 19 "                   " }}{PARA 260 "" 0 "" 
{TEXT -1 3 "   " }{XPPEDIT 18 0 "L[2];" "6#&%\"LG6#\"\"#" }{TEXT -1 3 
":  " }{XPPEDIT 18 0 "1/3*x-5/3;" "6#,&*(\"\"\"F%\"\"$!\"\"%\"xGF%F%*&
\"\"&F%F&F'F'" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "1/2*y-1/2;" "6#,&*(
\"\"\"F%\"\"#!\"\"%\"yGF%F%*&F%F%F&F'F'" }{TEXT -1 4 "  = " }{XPPEDIT 
18 0 "z-4;" "6#,&%\"zG\"\"\"\"\"%!\"\"" }{TEXT -1 17 "                \+
 " }}{PARA 0 "" 0 "" {TEXT 259 57 "Determine whether lines intersect, \+
are skew, or parallel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 215 "First, note that the direction vector of the first \+
line is <1,2,3>, and for the second line it's <3,2,1>.  Since these ve
ctors are not parallel (i.e. multiples of each other), the lines eithe
r intersect or are skew." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 1 " " }{TEXT 263 10 "Method 1) " }{TEXT -1 2 " \"" }
{TEXT 262 5 "solve" }{TEXT -1 174 "\" command\": we are really asking \+
for whether four planes intersect, since the symmetric form of a line \+
can be interpreted as giving the line as an intersection of two planes
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "solve(\{x-2=(y+1)/2,x-2
=(z-3)/3,\n       (x-5)/3=(y-1)/2,(x-5)/3=z-4\},\{x,y,z\});" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 9 "Method 2)" }
{TEXT -1 118 " Write the 4 linear equations above as a system, do elem
entary row operations to find the solution.  We can write the " }}
{PARA 0 "" 0 "" {TEXT -1 12 "equations as" }}{PARA 265 "" 0 "" 
{XPPEDIT 18 0 "2*x-y = 5;" "6#/,&*&\"\"#\"\"\"%\"xGF'F'%\"yG!\"\"\"\"&
" }{TEXT -1 0 "" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "3*x-z = 3;" "6#/,&
*&\"\"$\"\"\"%\"xGF'F'%\"zG!\"\"F&" }{TEXT -1 0 "" }}{PARA 263 "" 0 "
" {XPPEDIT 18 0 "2*x-3*y = 7;" "6#/,&*&\"\"#\"\"\"%\"xGF'F'*&\"\"$F'%
\"yGF'!\"\"\"\"(" }{TEXT -1 0 "" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "x-
3*z = -7;" "6#/,&%\"xG\"\"\"*&\"\"$F&%\"zGF&!\"\"!\"(" }{TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 51 "Which, when we write it synthetically g
ives us the " }{TEXT 269 10 "augmented " }{TEXT -1 103 "matrix A  (The
 matrix of coefficients is augmented with the column of right hand sid
es, hence the name " }{TEXT 270 9 "augmented" }{TEXT -1 2 "):" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "with(linalg):  #load matrix
 and linear algebra package\nA:=matrix(4,4,[2,-1,0,5,3,0,-1,3,2,-3,0,7
,1,0,-3,-7]);\nrref(A);  #compute the reduced row echelon form of the \+
matrix" }}}{PARA 0 "" 0 "" {TEXT -1 61 "We can read off the solution x
=2, y=-1, z=3 from the rref(A)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 266 9 "Method 3)" }{TEXT -1 109 "  Convert the s
ymmetric forms of the lines into a parametric equations, and use \"sol
ve\" again.  For example, " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "r1:=t-><2,-1,3>+t*<1,2,3>;\nr2:=s-><5,1,4>+s*<3,2,1>;\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "solve(\{t+2=3*s+5,2*t-1=2*s+
1,3*t+3=s+4\},\{s,t\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "
r1(0)=r2(-1);  #get point and check answer" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 9 "Method 4)" }{TEXT -1 38 " Like \+
method (2), only to get s and t:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 47 "B:=matrix(3,3,[-3,1,3,-2,2,2,-1,3,1]);\nrref(B);" }}}{PARA 0 "
" 0 "" {TEXT -1 68 "i.e. s=-1 and t=0, from which you recover the poin
t of intersection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 268 9 "Example: " }{TEXT -1 63 "Planes, parametrically:  Conside
r the plane given implicitly by" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "x-
2*y+z = 4;" "6#/,(%\"xG\"\"\"*&\"\"#F&%\"yGF&!\"\"%\"zGF&\"\"%" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "This single equation corr
esponds to a very short augmented matrix " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "C:=matrix(1,4,[1,-2,1,4]);" }}}{PARA 0 "" 0 "" {TEXT 
-1 129 "This matrix is already in reduced row echelon form, and we can
 backsolve: z=t, y=s, x=4+2*s-t.  We can write this in vector form:" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "F:=(s,t)-><4,0,0>+s*<2,1,0>
+t*<-1,0,1>; #F is the position vector\n    #of points on the plane" }
}}{PARA 0 "" 0 "" {TEXT -1 175 "Can you explain, with your understandi
ng of what vector addition and scalar multiplication, why this collect
ion of points gives a plane?  We can check with a plotting command:" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(plots);  #load the plo
tting package\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "?plot3d;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot3d([4+2*s-t,s,t],s=
0..1,t=0..1,axes=boxed);" }}}{PARA 0 "" 0 "" {TEXT -1 62 "Would you li
ke to try one of your homework problems from 11.5?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "24 2" 70 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }