Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.3
Use the appropriate formula from Table 1.5.1 to calculate the area of the parallelogram whose vertices are the four points P:4,13, Q:12,29, R:16,57, and S:8,41.
Solution
Mathematical Solution
Figure 1.5.3(a) shows the parallelogram formed by the points P, Q, R, and S. If P, Q, and S are position vectors to the points P, Q, and S, then two adjacent edges of the parallelogram are described by the vectors
A=Q−P = 12290−4130 = 8160
and
B=S−P = 8410−4130 = 4280
Figure 1.5.3(a) Parallelogram PQRS
The area of the parallelogram is then the magnitude of
A×B= |ijk81604280| = 008⋅28−4⋅16 = 00160
easily seen to be 160.
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
4,13,0→assign to a nameP
Enter Q as per Table 1.1.1.
Context Panel: Assign to a Name≻Q
12,29,0→assign to a nameQ
Enter S as per Table 1.1.1.
Context Panel: Assign to a Name≻S
8,41,0→assign to a nameS
By subtraction, obtain the vectors A and B along the edges of the parallelogram
Context Panel: Assign Name
A=Q−P→assign
B=S−P→assign
Obtain the area of the parallelogram as 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 = 160
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the position vectors P, Q, and S.
P,Q,S≔4,13,0,12,29,0,8,41,0:
Obtain vectors A and B along the edges of the parallelogram.
A,B≔Q−P,S−P:
Compute the norm of the cross product of A and B.
NormCrossProductA,B = 160
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