Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.4
Use the appropriate formula from Table 1.5.1 to calculate the area of the triangle whose vertices are the three points P:1,2,3, Q:−5,3,2, and R:7,−5,4.
Solution
Mathematical Solution
Figure 1.5.4(a) shows the triangle formed by the points P, Q, and R. If P, Q, and R are the respective position vectors to these points, then two adjacent edges of the triangle are described by the vectors
A=Q−P = −532−123 = −61−1
B=R−P = 7−54−123 = 6−71
use plots,VectorCalculus in module() local p1,p2,p3,P,Q,R,E1,E2; P,Q,R := <1,2,3>,<-5,3,2>,<7,-5,4>: E1:=RootedVector(root=[1,2,3],Q-P); E2:=RootedVector(root=[1,2,3],R-P); p1:=spacecurve([[1,2,3],[-5,3,2],[7,-5,4],[1,2,3]],color=red); p2:=PlotVector([E1,E2],color=[green,gold],width=.3); p3:=display(p1,p2,scaling=constrained,axes=box,labels=[x,y,z],tickmarks=[6,8,3],orientation=[-175,65,0],lightmodel=none,glossiness=0); print(p3); end module: end use:
Figure 1.5.4(a) Triangle PQR
The vectors A (in green) and B (in gold) emanate from point P; point Q is the tip of A and point R, the tip of B. The area of the triangle is half the magnitude of
A×B= |ijk−61−16−71| = −6036
or 637/2=337.
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
By subtraction, obtain the vectors A and B along the edges of the triangle
Context Panel: Assign Name
A=Q−P→assign
B=R−P→assign
Obtain the area of the triangle as half the norm of the cross product of A and B
Keyboard the norm bars.
Common Symbols palette: Cross-product operator
Context Panel: Evaluate and Display Inline
A×B/2 = 337
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the position vectors P, Q, and R.
P,Q,R≔1,2,3,−5,3,2,7,−5,4:
Obtain vectors A and B along the edges of the triangle.
A,B≔Q−P,R−P:
Compute half the norm of the cross product of A and B
Use the CrossProduct and Norm commands.
NormCrossProductA,B/2 = 337
<< Previous Example Section 1.5 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
