Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.10
Using the formulas in Table 1.5.1 for reciprocal vectors, obtain V1,V2,V3, the set of vectors reciprocal to the vectors U1=A, U2=B, U3=C, where A, B, and C are given in Example 1.5.1.
Solution
Mathematical Solution
Begin by computing the box product
λ=U1U2U3=U1·U2×U3 = |3−2425−4576| = 194
Next, obtain the following three cross products.
U2×U3= |ijk25−4576| = 58−32−11
U3×U1=|ijk5763−24| = 40−2−31
U1×U2=|ijk3−2425−4| = −122019
Finally, divide each of the three cross products by λ.
V1=1194 58−32−11
V2=119440−2−31
V3=1194−122019
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Enter the vectors U1,U2,U3
Enter A as per Table 1.1.1.
Context Panel: Assign to a Name≻U[1]
3,−2,4→assign to a nameU1
Enter B as per Table 1.1.1.
Context Panel: Assign to a Name≻U[2]
2,5,−4→assign to a nameU2
Enter C as per Table 1.1.1.
Context Panel: Assign to a Name≻U[3]
5,7,6→assign to a nameU3
Compute λ=U1U2U3
Write a sequence of the names of the three vectors. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Triple Scalar Product
Context Panel: Assign to a Name≻lambda
U1,U2,U3 = →scalar triple product194→assign to a nameλ
Obtain the reciprocal vectors as per the formulas in Table 1.5.1
Common Symbols palette: Cross-product operator
Context Panel: Assign Name
V1=U2×U3λ→assign
V2=U3×U1λ→assign
V3=U1×U2λ→assign
Display the reciprocal vectors
V1,V2,V3
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the vectors U1, U2, and U3.
U1,U2,U3≔3,−2,4,2,5,−4,5,7,6:
Apply the BoxProduct command.
λ≔BoxProductU1,U2,U3:
Apply the CrossProduct command to obtain the reciprocal vectors as per the formulas in Table 1.5.1
V1≔CrossProductU2,U3/λ:V2≔CrossProductU3,U1/λ:V3≔CrossProductU1,U2/λ:
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