Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
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Example 1.7.10
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Obtain an equation for L, the line that is parallel to and , the planes
and
and intersects and , the lines
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Solution
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Mathematical Solution
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For line to be parallel to planes and , it has to have the same direction as the line of intersection of and .
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This can be seen in Figure 1.7.10(a) where line , drawn in green, is parallel to the intersection of planes and . (Rotate the figure to see this is so.)
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Let V be the direction of . Then a vector from to is given by . This vector must be parallel to V. Hence, solve for .
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can then be given by .
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use plots in
module()
local P,Q,S,V,L1,L2,q,L,p1,p2,p3,p4,p5;
P:=2*x-7*y+9*z=12:
Q:=3*x+5*y-8*z=1:
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V:=map(coeff,eval(<x,y,z>,S),z):
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L1,L2:=<2*u+1,2-1*u,3+u>,<2+v,-1-3*v,2+2*v>:
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q:=solve(Equate(L1-L2,k*V),{u,v,k}):
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L:=eval(L1,q)-s*eval(k*V,q):
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p1:=implicitplot3d([P,Q],x=-2..5,y=-3..5,z=-1..5,style=surface,color=[red,gold]):
p2:=spacecurve(L1,u=-1..2,color=black,thickness=3,numpoints=2):
p3:=spacecurve(L2,v=-1..1,color=blue,thickness=3,numpoints=2):
p4:=spacecurve(L,s=0..1,color=green,thickness=5,numpoints=2):
p5:=display([p1,p2,p3,p4],scaling=constrained,axes=none,orientation=[-175,75,-75],lightmodel=none):
print(p5);
end module:
end use:
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Figure 1.7.10(a) Planes and (red and gold, respectively); lines and (black and blue, respectively); line (green)
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The vector V can be obtained as , where and are normals for planes and , respectively. It can also be obtained via the "Lines & Planes" tools in the Student MultivariateCalculus package.
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Maple Solution - Interactive
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Define lines and as position vectors and
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Context Panel: Assign to a Name≻L[1]
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Context Panel: Assign to a Name≻L[2]
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Define and , normals to planes and , respectively
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Context Panel: Assign to a Name≻N[P]
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Context Panel: Assign to a Name≻N[Q]
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Obtain V, the direction vector for the line of intersection of planes and
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Write for the direction of line .
Context Panel: Assign Name
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Solve the equations for
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Write and press the Enter key.
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Context Panel: Conversions≻To List
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Context Panel: Solve≻Solve
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Context Panel: Assign to a Name≻
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Obtain line
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Expression palette: Evaluation template
Evaluate at .
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Maple Solution - Coded
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The following calculations obtain line via the "Lines & Planes" tools of the Student MultivariateCalculus package. Collinearity of the direction vectors for the line of intersection of planes and and the line through generic points on lines and can be established either by showing their cross product is the zero vector, or by showing that the vectors are proportional. The first approach is taken in the Interactive Solution above; the second, in the Coded Solution below.
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Install the Student MultivariateCalculus package.
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Use the Plane command to define plane .
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Use the Plane command to define plane .
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Use the GetIntersection command to obtain the line of intersection of planes and .
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Use the GetDirection command to get the direction vector for the line of intersection.
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Define A, the generic point on line .
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Define B, the generic point on line .
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Use the Line command to obtain the line from A to B.
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Use the GetDirection command to obtain W, the direction vector for the line through A and B.
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Use the Equate command to equate corresponding components of W and a multiple of V.
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Use the solve command to solve for .
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Use the eval command to evaluate the generic point A at the solution .
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Write the vector form of the line with direction V that intersects lines and .
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