Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.1
If A=3 i−2 j+4 j, B=2 i+5 j−4 k, and C=5 i+7 j+6 k,
Compute ABC, the Triple Scalar (or Box) Product A·B×C.
Verify the identity A·B×C=A×B·C for the Triple Scalar Product.
Solution
Mathematical Solution
Part (a)
ABC= A·B×C = |3−2425−4576| = 194
Indeed,
A·B×C = 3−24·58−32−11 = 174+64−44=194
The cross product is computed first, resulting in the dot product between two vectors. If the dot product were done first, producing a scalar, the second operation would be the cross product between that scalar and a vector, an operation not defined. Hence, the notation A·B×C implies that the cross product must be done first.
Part (b)
A×B·C = C·A×B = |5763−2425−4| = −|3−2457625−4| = |3−2425−4576| = A·B×C
The dot product between two vectors commutes, giving the first equality. The interchange of two adjacent rows in a determinant negates the determinant. This is done twice, first interchanging rows 1 and 2, then interchanging rows 2 and 3. This final determinant now expresses ABC = A·B×C.
Maple Solution - Interactive
Initialization
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Enter A as per Table 1.1.1.
Context Panel: Assign to a Name≻A
3,−2,4→assign to a nameA
Enter B as per Table 1.1.1.
Context Panel: Assign to a Name≻B
2,5,−4→assign to a nameB
Enter C as per Table 1.1.1.
Context Panel: Assign to a Name≻C
5,7,6→assign to a nameC
Solution via Context Panel
Write a sequence of the names of the three vectors. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Triple Scalar Product
A,B,C = 3−24,25−4,576→scalar triple product194
Solution from first principles
Common Symbols palette: Dot-product and cross-product operators Note the (essential) use of parentheses.
Context Panel: Evaluate and Display Inline
A·B×C = 194
Obtain the Box Product CAB
C,A,B = 576,3−24,25−4→scalar triple product194
Common Symbols palette: Dot-product and cross-product operators
A×B·C = 194
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the vectors A, B, and C.
A,B,C≔3,−2,4,2,5,−4,5,7,6:
Apply the BoxProduct command.
BoxProductA,B,C = 194
Use the DotProduct and CrossProduct commands to obtain ABC = A·B×C.
DotProductA,CrossProductB,C = 194
In the Student MultivariateCalculus package, the commands BoxProduct and TripleScalarProduct are equivalent. Either form of the command can be used.
BoxProductC,A,B = 194
Use the DotProduct and CrossProduct commands to obtain A×B·C.
DotProductCrossProductA,B,C = 194
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