Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
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.
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
P,Q,R := <1,2,3>,<-5,3,2>,<7,-5,4>:
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
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
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
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 = 3⁢37
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the position vectors P, Q, and R.
Obtain vectors A and B along the edges of the triangle.
Compute half the norm of the cross product of A and B
Use the CrossProduct and Norm commands.
NormCrossProductA,B/2 = 3⁢37
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