Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.2
For the vectors A, B, and C of Example 1.5.1, and D=4 i+3 j−2 k,
Verify the identity A×B×C×D=ACDB−BCDA.
Verify the identity A×B×C×D=ABDC−ABCD.
Solution
Maple Solution - Interactive
Initialization
Make D, the differentiation operator, useable as a variable name
Invoke the local command. Press the Enter key.
local D≔D:
D
Define the vectors A, B, C, and D
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
Enter D as per Table 1.1.1.
Context Panel: Assign to a Name≻D
4,3,−2→assign to a nameD
Part (a)
Evaluate the left-hand side of the identity
Common Symbols palette: Cross-product operator
Context Panel: Evaluate and Display Inline
A×B×C×D = −906−764232
Evaluate the right-hand side of the identity
Common Symbols palette: Cross-product and dot-product operators
A·C×D⋅B−B·C×D⋅A = −906−764232
Part (b)
A·B×D⋅C−A·B×C⋅D = −906−764232
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the vectors A, B, C, and D. D must still be declared local, as per the previous section.
A,B,C,D≔3,−2,4,2,5,−4,5,7,6,4,3,−2:
CrossProductCrossProductA,B,CrossProductC,D = −906−764232
BoxProductA,C,D⋅B−BoxProductB,C,D⋅A = −906−764232
Alternatively, use first principles to evaluate the right-hand side
DotProductA,CrossProductC,D⋅B−DotProductB,CrossProductC,D⋅A = −906−764232
BoxProductA,B,D⋅C−BoxProductA,B,C⋅D = −906−764232
DotProductA,CrossProductB,D⋅C−DotProductA,CrossProductB,C⋅D = −906−764232
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