restart:
with(Maplets[Tools]): with(Maplets[Elements]): with(plots): with(plottools):
StartEngine();
randomize():
randa:=rand(-3..3):
rand2:=rand(0..1):
old:=[
[1,1],
[3,4], [4,3], [6,8], [8,6], [9,12], [12,9], [12,16], [16,12], [15,20], [20,15],
[5,12], [10,24], [15,36], [36,15], [20,48], [48,20], [25,60], [60,25],
[7,24], [24,7], [14,48], [48,14], [8,15], [15,8], [16,30], [30,16],
[12,35], [35,12], [24,70], [70,24], [16,63], [63,16],
[20,21], [21,20], [40,42], [42,40], [28,45], [45,28], [33,56], [56,33],
[36,77], [77,36], [39,80], [80,39], [48,55], [55,48], [65,72], [72,65],
NULL
]:
vecs:=[
[1,2,2], [3,4,12], [6,8,24], [9,12,36],
NULL
]:
randvec:=rand(1..nops(vecs)):
ask:=proc()
global pA,pB,pC,angleA,angleB,angleC,Area,VecAB, VecAC, MagAB, MagAC, DotprodA, EqA, VecBA, VecBC, MagBA, MagBC, DotprodB, EqB, VecCA, VecCB, MagCA, MagCB, DotprodC, EqC, crossX, crossY, crossZ,crossVEC, crossMAG, p1, p2, trianglepic;
local vAB, vAC, xmax, xmin, ymax, ymin, zmax, zmin, xwidth, ywidth, zwidth;
pA:=[randa(),randa(),randa()];
vAB:=combinat[randperm](zip(`*`,[(-1)^rand2(),(-1)^rand2(),(-1)^rand2()],vecs[randvec()]));
vAC:=combinat[randperm](zip(`*`,[(-1)^rand2(),(-1)^rand2(),(-1)^rand2()],vecs[randvec()]));
crossX:=vAB[2]*vAC[3]-vAB[3]*vAC[2];
crossY:=-(vAB[1]*vAC[3]-vAB[3]*vAC[1]);
crossZ:=(vAB[1]*vAC[2]-vAB[2]*vAC[1]);
while crossX=0 and crossY=0 and crossZ=0 do
vAC:=combinat[randperm](zip(`*`,[(-1)^rand2(),(-1)^rand2(),(-1)^rand2()],vecs[randvec()]));
crossX:=vAB[2]*vAC[3]-vAB[3]*vAC[2];
crossY:=-(vAB[1]*vAC[3]-vAB[3]*vAC[1]);
crossZ:=(vAB[1]*vAC[2]-vAB[2]*vAC[1]);
end do;
pB:=pA+vAB;
pC:=pA+vAC;
Set(point1=pA);
Set(point2=pB);
Set(point3=pC);
Set(vectorAB="");
Set(vectorAC="");
Set(vectorBC="");
Set(lengthAB="");
Set(lengthAC="");
Set(lengthBC="");
Set(ABdotAC="");
Set(BAdotBC="");
Set(CAdotCB="");
Set(anglea="");
Set(angleb="");
Set(anglec="");
Set(cross="");
Set(area="");
#solve angle A
VecAB:=pB-pA;
VecAC:=pC-pA;
MagAB:=sqrt(VecAB[1]^2+VecAB[2]^2+VecAB[3]^2):
MagAC:=sqrt(VecAC[1]^2+VecAC[2]^2+VecAC[3]^2):
DotprodA:=VecAB[1]*VecAC[1]+VecAB[2]*VecAC[2]+VecAB[3]*VecAC[3]:
EqA:=DotprodA=MagAB*MagAC*cos(thetaA):
angleA:=evalf(solve(EqA,thetaA)*180/Pi):
#solve angle B
VecBA:=pA-pB:
VecBC:=pC-pB:
MagBA:=sqrt(VecBA[1]^2+VecBA[2]^2+VecBA[3]^2):
MagBC:=sqrt(VecBC[1]^2+VecBC[2]^2+VecBC[3]^2):
DotprodB:=VecBA[1]*VecBC[1]+VecBA[2]*VecBC[2]+VecBA[3]*VecBC[3]:
EqB:=DotprodB=MagBA*MagBC*cos(thetaB):
angleB:=evalf(solve(EqB,thetaB)*180/Pi):
#solve angle C
VecCA:=pA-pC:
VecCB:=pB-pC:
MagCA:=sqrt(VecCA[1]^2+VecCA[2]^2+VecCA[3]^2):
MagCB:=sqrt(VecCB[1]^2+VecCB[2]^2+VecCB[3]^2):
DotprodC:=VecCA[1]*VecCB[1]+VecCA[2]*VecCB[2]+VecCA[3]*VecCB[3]:
EqC:=DotprodC=MagCA*MagCB*cos(thetaC):
angleC:=evalf(solve(EqC,thetaC)*180/Pi):
#doing the cross product
crossX:=VecAB[2]*VecBC[3]-VecAB[3]*VecBC[2];
crossY:=-(VecAB[1]*VecBC[3]-VecAB[3]*VecBC[1]);
crossZ:=(VecAB[1]*VecBC[2]-VecAB[2]*VecBC[1]);
crossVEC:=[crossX,crossY,crossZ];
crossMAG:=sqrt(crossX^2+crossY^2+crossZ^2);
Area:=crossMAG/2;
#plot
xmax:=max(pA[1],pB[1],pC[1]): xmin:=min(pA[1],pB[1],pC[1]);
ymax:=max(pA[2],pB[2],pC[2]): ymin:=min(pA[2],pB[2],pC[2]);
zmax:=max(pA[3],pB[3],pC[3]): zmin:=min(pA[3],pB[3],pC[3]);
xwidth:=xmax-xmin;
ywidth:=ymax-ymin;
zwidth:=zmax-zmin;
p1:=spacecurve([pA,pB,pC,pA], color=red);
p2:=textplot3d({[pA[1],pA[2],pA[3]," A"],
[pB[1],pB[2],pB[3]," B"],
[pC[1],pC[2],pC[3]," C"]},
font=[TIMES,BOLD,16], align=RIGHT, color=blue);
trianglepic:=display(p1,p2,
view=[(min(pA[1],pB[1],pC[1])-.1*xwidth)..(max(pA[1],pB[1],pC[1])+.1*xwidth),
(min(pA[2],pB[2],pC[2])-.1*ywidth)..(max(pA[2],pB[2],pC[2])+.1*ywidth),
(min(pA[3],pB[3],pC[3])-.1*zwidth)..(max(pA[3],pB[3],pC[3])+.1*zwidth)],
orientation=[30,75], axes=normal, tickmarks=[0,0,0], labels=[x,y,z], thickness=2);
Set(pic=trianglepic);
Set(reply="");
end proc:
check_vectorAB:=proc()
global VecAB;
local userAB;
userAB:=Get(vectorAB::anything):
if evalf(abs(userAB[1]-VecAB[1])<=.01*abs(VecAB[1])) and
evalf(abs(userAB[2]-VecAB[2])<=.01*abs(VecAB[2])) and
evalf(abs(userAB[3]-VecAB[3])<=.01*abs(VecAB[3]))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_vectorAB:=proc()
global VecAB;
Set(vectorAB=VecAB);
Set(reply="Try doing it yourself next time")
end proc:
check_vectorAC:=proc()
global VecAC;
local userAC;
userAC:=Get(vectorAC::anything):
if evalf(abs(userAC[1]-VecAC[1])<=.01*abs(VecAC[1])) and
evalf(abs(userAC[2]-VecAC[2])<=.01*abs(VecAC[2])) and
evalf(abs(userAC[3]-VecAC[3])<=.01*abs(VecAC[3]))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_vectorAC:=proc()
global VecAC;
Set(vectorAC=VecAC);
Set(reply="Try doing it yourself next time")
end proc:
check_vectorBC:=proc()
global VecBC;
local userBC;
userBC:=Get(vectorBC::anything):
if evalf(abs(userBC[1]-VecBC[1])<=.01*abs(VecBC[1])) and
evalf(abs(userBC[2]-VecBC[2])<=.01*abs(VecBC[2])) and
evalf(abs(userBC[3]-VecBC[3])<=.01*abs(VecBC[3]))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_vectorBC:=proc()
global VecBC;
Set(vectorBC=VecBC);
Set(reply="Try doing it yourself next time")
end proc:
check_lengthAB:=proc()
global MagAB;
local userLAB;
userLAB:=Get(lengthAB::anything):
if evalf(abs(userLAB-MagAB)<.01*abs(MagAB))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_lengthAB:=proc()
global MagAB;
Set(lengthAB=MagAB);
Set(reply="Try doing it yourself next time")
end proc:
check_lengthAC:=proc()
global MagAC;
local userLAC;
userLAC:=Get(lengthAC::anything):
if evalf(abs(userLAC-MagAC)<.01*abs(MagAC))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_lengthAC:=proc()
global MagAC;
Set(lengthAC=MagAC);
Set(reply="Try doing it yourself next time")
end proc:
check_lengthBC:=proc()
global MagBC;
local userLBC;
userLBC:=Get(lengthBC::anything):
if evalf(abs(userLBC-MagBC)<.01*abs(MagBC))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_lengthBC:=proc()
global MagBC;
Set(lengthBC=MagBC);
Set(reply="Try doing it yourself next time")
end proc:
check_ABdotAC:=proc()
global DotprodA;
local userABAC;
userABAC:=Get(ABdotAC::anything):
if evalf(abs(userABAC-DotprodA)<=.01*abs(DotprodA))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_ABdotAC:=proc()
global DotprodA;
Set(ABdotAC=DotprodA);
Set(reply="Try doing it yourself next time")
end proc:
check_BAdotBC:=proc()
global DotprodB;
local userBABC;
userBABC:=Get(BAdotBC::anything):
if evalf(abs(userBABC-DotprodB)<=.01*abs(DotprodB))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_BAdotBC:=proc()
global DotprodB;
Set(BAdotBC=DotprodB);
Set(reply="Try doing it yourself next time")
end proc:
check_CAdotCB:=proc()
global DotprodC;
local userCACB;
userCACB:=Get(CAdotCB::anything):
if evalf(abs(userCACB-DotprodC)<=.01*abs(DotprodC))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_CAdotCB:=proc()
global DotprodC;
Set(CAdotCB=DotprodC);
Set(reply="Try doing it yourself next time")
end proc:
JSFH
check_anglea:=proc()
global pA,pB,pC,angleA,angleB,angleC,Area;
local useranglea;
useranglea:=Get(anglea::anything):
if evalf(abs(useranglea-angleA)<.01*abs(angleA))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_anglea:=proc()
global pA,pB,pC,angleA,angleB,angleC,Area;
Set(anglea=angleA);
Set(reply="Try doing it yourself next time")
end proc:
check_angleb:=proc()
global pA,pB,pC,angleA,angleB,angleC,Area;
local userangleb;
userangleb:=Get(angleb::anything):
if evalf(abs(userangleb-angleB)<.01*abs(angleB))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_angleb:=proc()
global pA,pB,pC,angleA,angleB,angleC,Area;
Set(angleb=angleB);
Set(reply="Try doing it yourself next time")
end proc:
check_anglec:=proc()
global pA,pB,pC,angleA,angleB,angleC,Area;
local useranglec;
useranglec:=Get(anglec::anything):
if evalf(abs(useranglec-angleC)<.01*abs(angleC))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_anglec:=proc()
global pA,pB,pC,angleA,angleB,angleC,Area;
Set(anglec=angleC);
Set(reply="Try doing it yourself next time")
end proc:
check_area:=proc()
global pA,pB,pC,angleA,angleB,angleC,Area;
local userarea;
userarea:=Get(area::anything):
if evalf(abs(userarea-Area)<.01*abs(Area))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_area:=proc()
global pA,pB,pC,angleA,angleB,angleC,Area;
Set(area=Area);
Set(reply="Try doing it yourself next time")
end proc:
check_cross:=proc()
global crossVEC;
local usercross;
usercross:=Get(cross::anything):
if evalf(abs(usercross[1]-crossVEC[1])<=.01*abs(crossVEC[1])) and
evalf(abs(usercross[2]-crossVEC[2])<=.01*abs(crossVEC[2])) and
evalf(abs(usercross[3]-crossVEC[3])<=.01*abs(crossVEC[3]))
then Set('reply'="Correct!")
else Set('reply'="Incorrect. Try again.")
end if;
end proc:
show_cross:=proc()
global crossVEC;
Set(cross=crossVEC);
Set(reply="Try doing it yourself next time")
end proc:
hint_vectors:=proc()
Set(reply="A vector is the difference between its two endpoints. PQ = Q - P Remember QP = - PQ")
end proc:
hint_mags:=proc()
Set(reply="Use the Pythagorean Thereom.")
end proc:
hint_dots:=proc()
Set(reply="[a1,a2,a3] . [b1,b2,b3] = a1*b1+a2*b2+a3*b3 ... Remember AB is not the same as BA")
end proc:
hint_angles:=proc()
Set(reply="Use the dot product. cos(alpha) = (AB . AC) / (|AB| |AC|)")
end proc:
hint_area:=proc()
Set(reply="The area is half of the magnitude of the cross product of two adjacent vectors.")
end proc:
hint_cross:=proc()
Set(reply="Compute a 3x3 determinant. (Row 1: i, j, k. Row 2: AB. Row 3: AC.)")
end proc:
Triangle:=Maplet(onstartup = RunWindow(MAIN),
Font[helv10]("helvetica",10),
Window[MAIN]('title'="Triangles in 3-Space",
[
[ Button("New Problem", Evaluate('function' = "ask"), background=magenta),
HorizontalGlue(),
Button("Quit",Shutdown(), background=pink)
],
[ inset=0, spacing=0,
[ valign=none, halign=none, border=true,
caption="Consider the Triangle with Vertices:",
[ halign=none, inset=0,
"A =",
TextField[point1](width=9, editable=false, background=white),
HorizontalGlue(),
"B =",
TextField[point2](width=9, editable=false, background=white)
],
[ halign=none, inset=0,
HorizontalGlue(),
"C =",
TextField[point3](width=9, editable=false, background=white),
HorizontalGlue()
],
[ halign=none,
Plotter[pic](width=250,height=250)
]
],
[ inset=0, spacing=0,
BoxColumn( border=true,
caption="Find the Side Vectors: Answer in the form: [x,y,z]",
[ halign=none, inset=0,
"AB =", TextField[vectorAB]('width'=8),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_vectorAB"), background=green),
Button("Show", Evaluate('function' = "show_vectorAB"), background=yellow),
Button("Hint", Evaluate('function' = "hint_vectors"), background=yellow)
],
[ halign=none, inset=0,
"AC =", TextField[vectorAC]('width'=8),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_vectorAC"), background=green),
Button("Show", Evaluate('function' = "show_vectorAC"), background=yellow),
Button("Hint", Evaluate('function' = "hint_vectors"), background=yellow)
],
[ halign=none, inset=0,
"BC =", TextField[vectorBC]('width'=8),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_vectorBC"), background=green),
Button("Show", Evaluate('function' = "show_vectorBC"), background=yellow),
Button("Hint", Evaluate('function' = "hint_vectors"), background=yellow)
]
),
BoxColumn( border=true,
caption="Compute each Dot Product:",
[ halign=none, inset=0,
"AB . AC =", TextField[ABdotAC]('width'=7),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_ABdotAC"), background=green),
Button("Show", Evaluate('function' = "show_ABdotAC"), background=yellow),
Button("Hint", Evaluate('function' = "hint_dots"), background=yellow)
],
[ halign=none, inset=0,
"BA . BC =", TextField[BAdotBC]('width'=7),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_BAdotBC"), background=green),
Button("Show", Evaluate('function' = "show_BAdotBC"), background=yellow),
Button("Hint", Evaluate('function' = "hint_dots"), background=yellow)
],
[ halign=none, inset=0,
"CA . CB =", TextField[CAdotCB]('width'=7),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_CAdotCB"), background=green),
Button("Show", Evaluate('function' = "show_CAdotCB"), background=yellow),
Button("Hint", Evaluate('function' = "hint_dots"), background=yellow)
]
),
BoxColumn( border=true,
caption="Find a Cross Product: Answer in the form: [x,y,z]",
[ halign=none, inset=0,
"AB x AC =", TextField[cross]('width'=9),
Button("Check", Evaluate('function' = "check_cross"), background=green),
Button("Show", Evaluate('function' = "show_cross"), background=yellow),
Button("Hint", Evaluate('function' = "hint_cross"), background=yellow)
]
)
],
[ inset=0, spacing=0,
BoxColumn( border=true,
caption="Find the Magnitude of each Side:",
[ halign=none, inset=0,
"|AB| =", TextField[lengthAB]('width'=7),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_lengthAB"), background=green),
Button("Show", Evaluate('function' = "show_lengthAB"), background=yellow),
Button("Hint", Evaluate('function' = "hint_mags"), background=yellow)
],
[ halign=none, inset=0,
"|AC| =", TextField[lengthAC]('width'=7),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_lengthAC"), background=green),
Button("Show", Evaluate('function' = "show_lengthAC"), background=yellow),
Button("Hint", Evaluate('function' = "hint_mags"), background=yellow)
],
[ halign=none, inset=0,
"|BC| =", TextField[lengthBC]('width'=7),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_lengthBC"), background=green),
Button("Show", Evaluate('function' = "show_lengthBC"), background=yellow),
Button("Hint", Evaluate('function' = "hint_mags"), background=yellow)
]
),
BoxColumn( border=true,
caption="Find Each Angle",
[ halign=none, inset=0,
"Angle A =", TextField[anglea]('width'=8),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_anglea"), background=green),
Button("Show", Evaluate('function' = "show_anglea"), background=yellow),
Button("Hint", Evaluate('function' = "hint_angles"), background=yellow)
],
[ halign=none, inset=0,
"Angle B =", TextField[angleb]('width'=8),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_angleb"), background=green),
Button("Show", Evaluate('function' = "show_angleb"), background=yellow),
Button("Hint", Evaluate('function' = "hint_angles"), background=yellow)
],
[ halign=none, inset=0,
"Angle C =", TextField[anglec]('width'=8),
HorizontalGlue(),
Button("Check", Evaluate('function' = "check_anglec"), background=green),
Button("Show", Evaluate('function' = "show_anglec"), background=yellow),
Button("Hint", Evaluate('function' = "hint_angles"), background=yellow)
]
),
BoxColumn( border=true,
caption="Find the Area",
[ halign=none, inset=0,
"Area = ", TextField[area]('width'=10),
Button("Check", Evaluate('function' = "check_area"), background=green),
Button("Show", Evaluate('function' = "show_area"), background=yellow),
Button("Hint", Evaluate('function' = "hint_area"), background=yellow)
]
)
]
],
[ TextField['reply']('width'=70, editable=false, background=white)
],
BoxRow('background'="#DDFFFF", 'inset'=0, 'spacing'=0,
Label("Programmer: John Horn", 'font'=helv10),
HorizontalGlue(),
Label("Copyright 2008, P. Yasskin", 'font'=helv10)
)
]
)
):
Maplets[Display]( Triangle );