Lines and Planes in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2KVEoJiM4NDc3O0YnLyUlc2l6ZUdRIzI0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGQC1GIzYnLUkjbW5HRiQ2KFEiM0YnL0YzUSMyMEYnRjVGO0Y+RkFGMi9GOUY3RjsvRj9RJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GP1Enbm9ybWFsRic=Distance from a Point to a Line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Problem LP-4</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Find the distance from the point P: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 to the line 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, where 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 and 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.
p1:=plots[spacecurve]([1+7*t,-1-9*t,2+5*t],t=-1..1.5,color=black,thickness=2,numpoints=2):
V:=VectorCalculus[RootedVector](root=[1,-1,2],<7,-9,5>):
p2:=VectorCalculus[PlotVector](V,color=red,width=.75):
L:=VectorCalculus[RootedVector](root=[1,-1,2],<1,2,-5>):
p3:=VectorCalculus[PlotVector](L,color=black,width=.75):
p4:=plot3d(-(35/23)*x+41/23-(40/23)*y,x=-8..12,y=-15..10,style=surface,transparency=.9,color=green):
plots[display](p1,p2,p3,p4,axes=none,scaling=constrained,view=-5..15,lightmodel=none,orientation=[-170,-60,-95]);Figure 1 Given line, vector V (red), vector L (black) from point A to point P
JSFHPoint P and the given line determine a unique plane containing both P and the line. In this plane, the vector L from point A to point P is projected onto and orthogonal to vector V. The length of the orthogonal component is the distance from point P to the line.JSFH
<Text-field style="Heading 1" layout="Heading 1">Solution by Task Template</Text-field>JSFHData for the problem:
Projection onto 1-D SubspaceVector to be projectedProject onto
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
if Do(%RadioButton1)=false then
Do(%ComboBox0(enabled)=false);
Do(%Label0(enabled)=false);
Do(%Label1(enabled)=false);
end if;
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
if Do(%RadioButton0)=false then
Do(%ComboBox0(enabled)=true);
Do(%Label0(enabled)=true);
Do(%Label1(enabled)=true);
end if;
end use;
[x\134,y],otheruse DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
if Do(%ComboBox0)=[x,y] then Do(%MathContainer3=NULL);end if;
end use;
use DocumentTools,plots, Student[LinearAlgebra] in
module()
local B,A,X,Y,Z,q1,q2,FIG, temp;
Do(B = %MathContainer0):
if Do(%RadioButton0) then Do(A = %MathContainer1);
if nops([A])=0 then Do(%Plot0 = display(textplot([0,0,"Provide a vector onto which to project"]),axes=none));
else
\011if Dimension(A)<>Dimension(B) then Do(%Plot0 = display(textplot([0,0,"Vectors should have the same dimension"]),axes=none));return;
\011elif nops(select(type,indets(A),name))<>0 or nops(select(type,indets(B),name))<>0 then Do(%Plot0 = display(textplot([0,0,"No graph can be drawn if vectors are not numeric"]),axes=none));return;
\011elif Dimension(A)=2 and Dimension(B)=2 then
\011FIG := ProjectionPlot(B,A, title="", complementoptions=[color=red], projectionoptions=[color=green], vectoroptions=[color=black]):
\011elif Dimension(A)=3 and Dimension(B)=3 then
\011FIG := ProjectionPlot(B,A, title="", complementoptions=[color=red], projectionoptions=[color=green], vectoroptions=[color=black], axes=box,labels=[x,y,z],orientation=[-55,50]):
\011end if;
\011Do(%Plot0 = FIG):
end if;
end if;
if Do(%RadioButton1) then
if Dimension(B)=3 and Do(%ComboBox0)=[x,y] then Do(%Plot0 = display(textplot([0,0,"Variables can't be [x,y] in 3D"]),axes=none));return;end;
if Dimension(B)=3 and Do(%ComboBox0)=other then Do(temp = %MathContainer3);
\011if nops([temp])=0 then Do(%MathContainer3 = MathML[Export]("???"));return;end if;
\011X:=temp[1];Y:=temp[2];Z:=temp[3];
\011Do(A = %MathContainer2);
FIG := ProjectionPlot(B,A, [X,Y,Z],title="", complementoptions=[color=red], projectionoptions=[color=green], vectoroptions=[color=black], axes=box,labels=[X,Y,Z],orientation=[-55,50]):
Do(%Plot0 = FIG);
return;end if;
\011if Do(%ComboBox0)=[x,y] then X:=x;Y:=y;end if;
\011if Do(%ComboBox0)=other then Do(temp=%MathContainer3);X:=temp[1];Y:=temp[2];end if;
\011Do(q1 = %MathContainer2);
\011if is(q1=NULL) then
\011Do(%MathContainer5=MathML[Export]("Provide equation of a line thru origin"));
else if type(q1,`=`) then q2:=lhs(q1)-rhs(q1); else q2:=q1;end if;
if verify(indets(q2),{X,Y},`subset`)=false then Do(%MathContainer4=MathML[Export]("Declared variables must match those in equation of line"));return;end if;
if eval(q2,[X=0,Y=0])<>0 then Do(%MathContainer5=MathML[Export]("The line does not pass thru origin"));
else
FIG := ProjectionPlot(B,q2,[X,Y], title="", complementoptions=[color=red], projectionoptions=[color=green], vectoroptions=[color=black]);
\011Do(%Plot0 = FIG);
end if;
end if;
end if;
end module;
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(%MathContainer0 = NULL);
Do(%MathContainer5 = NULL);
Do(%MathContainer1 = NULL);
Do(%MathContainer2 = NULL);
Do(%MathContainer4 = NULL);
Do(%MathContainer3 = NULL);
Do(%Plot0 = plots[display](plots[textplot]([0,0," "]),axes=none));
end use;
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 DocumentTools, Student[LinearAlgebra] in
module()
local B,A,BA,q1,q2,X,Y,temp,Temp;
Do(B = %MathContainer0):
if Do(%RadioButton0) then Do(A = %MathContainer1);
if nops([A])=0 then Do(%MathContainer4 = MathML[Export]("Provide a vector onto which to project"));
else
\011Temp := GetProperty(MathContainer5,value);
\011if StringTools:-Has(Temp,"P") then Do(%MathContainer5 = NULL);end if;
\011if Dimension(B)=Dimension(A) then
\011BA:=(B.A)/(A.A)*A;
\011Do(%MathContainer4 = BA);
\011else Do(%MathContainer4 = MathML[Export]("Vectors should have the same dimension"));return;end if;
end if;
end if;
if Do(%RadioButton1) then
if Dimension(B)=3 and Do(%ComboBox0)=[x,y] then Do(%MathContainer4 = MathML[Export]("Variables can't be [x,y] in 3D"));return;end if;
\011if Dimension(B)=3 and Do(%ComboBox0)=other then
\011Do(A = %MathContainer2);
\011BA := (B.A)/(A.A)*A;\011
\011Do(%MathContainer4 = BA);
\011return;
\011end if;
if Do(%ComboBox0)=[x,y] then X:=x;Y:=y;end if;
\011if Do(%ComboBox0)=other then Do(temp=%MathContainer3);X:=temp[1];Y:=temp[2];end if;
\011Do(q1 = %MathContainer2);
\011if is(q1=NULL) then
\011Do(%MathContainer4=MathML[Export]("Provide equation of a line thru origin"));
else if type(q1,`=`) then q2:=lhs(q1)-rhs(q1); else q2:=q1;end if;
if verify(indets(q2),{X,Y},`subset`)=false then Do(%MathContainer4=MathML[Export]("Declared coordinates must match variables in equation of line"));return;end if;
if eval(q2,[X=0,Y=0])<>0 then Do(%MathContainer4=MathML[Export]("The line does not pass thru origin"));
else
if has(q2,X) and has(q2,Y) then A:=<1,coeff(solve(q2,Y),X)>;BA:=(B.A)/(A.A)*A;Do(%MathContainer4 = BA);
elif has(q2,X) and not has(q2,Y) then BA:=<0,B[2]>;Do(%MathContainer4 = BA);
else BA:=<B[1],0>;Do(%MathContainer4 = BA); end if;
end if;
end if;
end if;
end module;
end use;
use DocumentTools in
module()
local temp;
try
GetProperty(MathContainer4,expression);
catch:
return;
end try;
Do(temp = %MathContainer4);
if nops([temp])=0 then return;end if;
Do(%MathContainer4 = evalf(temp));
end module;
end use;
use DocumentTools, Student[LinearAlgebra] in
module()
local A,B,BA,BpA,q1,q2,X,Y,temp;
Do(B = %MathContainer0):
if Do(%RadioButton0) then Do(A = %MathContainer1);
if nops([A])=0 then Do(%MathContainer5 = MathML[Export]("Provide a vector onto which to project"));
else
\011temp := GetProperty(MathContainer4,value);
\011if StringTools:-Has(temp,"P") then Do(%MathContainer4 = NULL);end if;
\011if Dimension(B)=Dimension(A) then
\011BA:=(B.A)/(A.A)*A;
\011BpA:=B-BA;
\011Do(%MathContainer5 = BpA);
\011else Do(%MathContainer5 = MathML[Export]("Vectors should have same dimension"));return;end if;
end if;
end if;
if Do(%RadioButton1) then
if Dimension(B)=3 and Do(%ComboBox0)=[x,y] then Do(%MathContainer5 = MathML[Export]("Variables can't be [x,y] in 3D"));return;end if;
if Dimension(B)=3 and Do(%ComboBox0)=other then
\011Do(A = %MathContainer2);
\011BA := (B.A)/(A.A)*A;
\011BpA := B-BA;
\011Do(%MathContainer5 = BpA);
\011return;
\011end if;
if Do(%ComboBox0)=[x,y] then X:=x;Y:=y;end if;
\011if Do(%ComboBox0)=other then Do(temp=%MathContainer3);X:=temp[1];Y:=temp[2];end if;
\011Do(q1 = %MathContainer2);
\011if is(q1=NULL) then
\011Do(%MathContainer5=MathML[Export]("Provide equation of a line thru origin"));
else if type(q1,`=`) then q2:=lhs(q1)-rhs(q1); else q2:=q1;end if;
if verify(indets(q2),{X,Y},`subset`)=false then Do(%MathContainer5=MathML[Export]("Declared coordinates must match variables in equation of line"));return;end if;
if eval(q2,[X=0,Y=0])<>0 then Do(%MathContainer5=MathML[Export]("The line does not pass thru origin"));
else
if has(q2,X) and has(q2,Y) then A:=<1,coeff(solve(q2,Y),X)>;BA:=(B.A)/(A.A)*A;BpA:=B-BA;Do(%MathContainer5 = BpA);
elif has(q2,X) and not has(q2,Y) then BpA:=<B[1],0>;Do(%MathContainer5 = BpA);
else BpA:=<0,B[2]>;Do(%MathContainer5 = BpA); end if;
end if;
end if;
end if;
end module;
end use;
use DocumentTools in
module()
local temp;
try
GetProperty(MathContainer5,expression);
catch:
return;
end try;
Do(temp = %MathContainer5);
if nops([temp])=0 then return;end if;
Do(%MathContainer5 = evalf(temp));
end module;
end use;
JSFHJSFHJSFHJSFHJSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Vector Solution from First Principles</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Obtain the length of the component of 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 orthogonal to V:
Write 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Context Menu: Assign NameLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The orthogonal component: 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Press the Enter key.Context Menu: Norm\342\211\273EuclideanLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
JSFH
<Text-field style="Heading 1" layout="Heading 1">Analytic Solution</Text-field>JSFHMinimize the distance from P to the line:
Write the generic vector from the line to P: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 the Enter key.
Context Menu: Norm\342\211\273EuclideanContext Menu: Simplify\342\211\273Assuming Real
Context Menu:
Differentiate\342\211\273With Respect To\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=Context Menu: Solve\342\211\273Solve
Expression palette: Evaluation template
Evaluate 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 at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJG1pbkYnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y/JSFH