Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
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Example 1.5.5
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Prove that the point S: does not lie in the plane determined by the points P, Q, and R given in Example 1.5.4.
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Solution
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Mathematical Solution
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The position vectors P, Q, R, and the "edge" vectors and are given in Example 1.5.4.
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In Figure 1.5.5(a), triangle PQR is delineated by the vectors A (in green) and B (in gold). The black vector is
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where S is the position vector to point S.
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If the vectors A, B, and C determine a parallelepiped with nonzero volume, point S cannot be in the plane of triangle PQR. If the volume of such a parallelepiped is zero, then point S lies in the plane of triangle PQR.
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>
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use plots,VectorCalculus in
module()
local p1,p2,p3,P,Q,R,S,E1,E2,V;
P,Q,R,S := <1,2,3>,<-5,3,2>,<7,-5,4>,<1,-1,4>:
E1:=RootedVector(root=[1,2,3],Q-P);
E2:=RootedVector(root=[1,2,3],R-P);
V:=RootedVector(root=[1,2,3],S-P);
p1:=spacecurve([[1,2,3],[-5,3,2],[7,-5,4],[1,2,3]],color=red);
p2:=PlotVector([E1,E2,V],color=[green,gold,black],width=.3);
p3:=display(p1,p2,scaling=constrained,axes=none,labels=[x,y,z],tickmarks=[6,8,3],orientation=[-135,50,0],lightmodel=none,glossiness=0);
print(p3);
end module:
end use:
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Figure 1.5.5(a) Vectors A, B, and C
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The volume of the parallelepiped is given by the absolute value of the triple scalar product
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The vectors A, B, and C determine a parallelepiped with nonzero volume. Hence, point S does not lie in the plane of triangle PQR.
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Maple Solution - Interactive
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Initialize
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Tools≻Load Package: Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Context Panel: Assign to a Name≻P
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Context Panel: Assign to a Name≻Q
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Context Panel: Assign to a Name≻R
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Context Panel: Assign to a Name≻S
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By subtraction, obtain the vectors A, B, and C
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Context Panel: Assign Name
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Context Panel: Assign Name
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Context Panel: Assign Name
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Obtain the triple scalar product
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Write a sequence of names of the three vectors.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Triple Scalar Product
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Obtain the triple scalar product from first principles
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Common Symbols palette: Dot- and cross-product operators
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Context Panel: Evaluate and Display Inline
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=
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Vectors A, B, and C determine a parallelepiped with nonzero volume. Hence point S, the tip of vector C, cannot lie in the plane of triangle PQR.
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Maple Solution - Coded
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Install the Student MultivariateCalculus package.
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Define the position vectors P, Q, R, and S.
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Obtain the vectors A, B, and C.
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Compute the Box Product
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Obtain from first principles
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=
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Vectors A, B, and C determine a parallelepiped with nonzero volume. Hence point S, the tip of vector C, cannot lie in the plane of triangle PQR.
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