Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.5
Prove that the point S:1,−1,4 does not lie in the plane determined by the points P, Q, and R given in Example 1.5.4.
Solution
Mathematical Solution
The position vectors P, Q, R, and the "edge" vectors A=Q−P and B=R−P are given in Example 1.5.4.
In Figure 1.5.5(a), triangle PQR is delineated by the vectors A (in green) and B (in gold). The black vector is
C=S−P = 1−14−123 = 0−31
where S is the position vector to point S.
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.
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:
Figure 1.5.5(a) Vectors A, B, and C
The volume of the parallelepiped is given by the absolute value of the triple scalar product
ABC = |−61−16−710−31| = 36≠0
The vectors A, B, and C determine a parallelepiped with nonzero volume. Hence, point S does not lie in the plane of triangle PQR.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Enter P as per Table 1.1.1.
Context Panel: Assign to a Name≻P
1,2,3→assign to a nameP
Enter Q as per Table 1.1.1.
Context Panel: Assign to a Name≻Q
−5,3,2→assign to a nameQ
Enter R as per Table 1.1.1.
Context Panel: Assign to a Name≻R
7,−5,4→assign to a nameR
Enter S as per Table 1.1.1.
Context Panel: Assign to a Name≻S
1,−1,4→assign to a nameS
By subtraction, obtain the vectors A, B, and C
Context Panel: Assign Name
A=Q−P→assign
B=R−P→assign
C=S−P→assign
Obtain the triple scalar product ABC
Write a sequence of names of the three vectors. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Triple Scalar Product
A,B,C = −61−1,6−71,0−31→scalar triple product36
Obtain the triple scalar product from first principles
Common Symbols palette: Dot- and cross-product operators
Context Panel: Evaluate and Display Inline
A·B×C = 36
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.
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the position vectors P, Q, R, and S.
P,Q,R,S≔1,2,3,−5,3,2,7,−5,4,1,−1,4:
Obtain the vectors A, B, and C.
A,B,C≔Q−P,R−P,S−P:
Compute the Box Product ABC=A·B×C
Apply the BoxProduct command.
BoxProductA,B,C = 36
Obtain ABC=A·B×C from first principles
Apply the DotProduct and CrossProduct commands.
DotProductA,CrossProductB,C = 36
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