Chapter 1: Vectors, Lines and Planes
Section 1.4: Cross Product
Example 1.4.3
For the vectors A, B, and C of Example 1.4.2, and V=2 i−5 j+3 k, verify the identity A×B·C×V=A·CA·VB·CB·V.
Solution
Mathematical Solution
The relevant calculations are as follows.
A×B= |ijk3−2457−6| = −163831 and C×V= |ijk4−3−52−53| = −34−22−14
A×B·C×V= 16⋅34−38⋅22−31⋅14= −726
A·C=12+6−20=−2A·V=6+10+12=28B·C=20−21+30=29B·V=10−35−18=−43} ⇒ A·CA·VB·CB·V = |−22829−43| = −726
Maple Solution - Interactive
Data entry
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
5,7,−6→assign to a nameB
Enter C as per Table 1.1.1.
Context Panel: Assign to a Name≻C
4,−3,−5→assign to a nameC
Enter V as per Table 1.1.1.
Context Panel: Assign to a Name≻V
2,−5,3→assign to a nameV
Left-hand Side
Right-hand Side
A×B·C×V = −726
|A·CA·VB·CB·V| = −726
It would be more natural to state the identity in terms of A, B, C, and D, but the symbol "D" is reserved in Maple for the differentiation operator. There are ways around this restriction, but the cure is worse than the illness.
To write the notation for the determinant, place single strokes around the matrix, fill in the entries of the matrix, then delete the brackets that denote "matrix". It is not essential to remove these brackets - Maple will still interpret the single vertical strokes as a call to the determinant function. But it looks nicer to delete the brackets.
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Install the Student LinearAlgebra package.
withStudent:-LinearAlgebra:
Define the vectors A, B, C, and V.
A,B,C,V≔3,−2,4,5,7,−6,4,−3,−5,2,−5,3:
Apply the DotProduct, CrossProduct, Determinant, and Matrix commands. The syntax used below in the Matrix command provides each row as a list-of-lists, with the sublists defining the rows of the matrix.
DotProductCrossProductA,B,CrossProductC,V = −726
DeterminantMatrixA·C,A·V,B·C,B·V = −726
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