Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Example 1.5.12
Solve the appropriate set of four equations in four unknowns to find V1,V2, the set of vectors reciprocal to U1=2 i−3 j, U2=3 i+4 j.
Solution
Mathematical Solution
Taking the Vk,k=1,2, as the vectors
a1a2,b1b2
the equations Ui·Vj=δij thereby generated are listed in Table 1.5.12(a). The solution of these equations is listed in Table 1.5.12(b). The resulting vectors Vk are listed in Table 1.5.12(c).
U1·V1=δ11
2a1−3 a2=1
U1·V2=δ12
2b1−3 b2=0
U2·V1=δ21
3a1+4a2=0
U2·V2=δ22
3b1+4b2=1
Table 1.5.12(a) The equations Ui·Vj=δij
a1=4/17
a2=−3/17
b1=3/17
b2=2/17
Table 1.5.12(b) Solution of equations in Table 1.5.12(a)
V1=117 4−3
V2=117 32
Table 1.5.12(c) Reciprocal vectors
Maple Solution - Interactive
Define the function δi,j={1i=j0i≠j
Expression palette: Piecewise template
Context Panel: Assign Function
δi,j=1i=j0i≠j→assign as functionδ
Define the vectors Uk and Vk, k=1,2
Context Panel: Assign Name U is the list of vectors U1 and U2.
U=2,−3,3,4→assign
Context Panel: Assign Name V is the list of vectors V1, and V2.
V=a1,a2,b1,b2→assign
Form the set of equations Ui·Vj=δij
Write Ui·Vj=δi,j and press the Enter key.
Context Panel: Sequence≻i≻ (See Figure 1.5.12(a).)
Context Panel: Conversions≻To List
Context Panel: Sequence≻j≻(See Figure 1.5.12(a).)
Context Panel: Join
Context Panel: Solve≻Solve
Context Panel: Assign to a Name≻S
Figure 1.5.12(a) Sequence dialog
Ui·Vj=δi,j
2−3,34i·a1a2,b1b2j=1i=j0i≠j
→sequence w.r.t. i
2−3·a1a2,b1b2j=11=j01≠j,34·a1a2,b1b2j=12=j02≠j
→to list
2−3·a1a2,b1b2j=11=j01≠j,34·a1a2,b1b2j=12=j02≠j
→sequence w.r.t. j
2a1−3a2=1,3a1+4a2=0,2b1−3b2=0,3b1+4b2=1
→list join
2a1−3a2=1,3a1+4a2=0,2b1−3b2=0,3b1+4b2=1
→solve
a1=417,a2=−317,b1=317,b2=217
→assign to a name
S
Evaluate each Vk at the solution S
Expression palette: Evaluation template
Context Panel: Evaluate and Display Inline
V1x=a|f(x)S = 417−317
V2x=a|f(x)S = 317217
The Context Panel option "Sequence" applies the seq command to generate a sequence of objects of similar structure. It cannot again be applied a second time to the resulting sequence of objects. Hence, the original sequence must be converted to a list, whereupon the Sequence option can again be applied. This results in a sequence of lists that can be combined into a single list via the Join option .
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Use the piecewise command to define the function
δi,j={1i=j0i≠j
δ≔i,j→piecewisei=j,1,i≠j,0:
Let U be the list of vectors U1 and U2.
U≔2,−3,3,4:
Let V be the list of vectors V1, and V2.
V≔a1,a2,b1,b2:
Use the seq command to generate the equations Ui·Vj=δij.
Apply the DotProduct command and assign the sequence of equations to the name q.
q≔seqseqDotProductUi,Vj=δi,j,i=1..2,j=1..2
q≔2a1−3a2=1,3a1+4a2=0,2b1−3b2=0,3b1+4b2=1
Apply the solve command to the set of equations Ui·Vj=δij
Assign the solution to the name S.
S≔solveq
S≔a1=417,a2=−317,b1=317,b2=217
Use the eval command to evaluate each vector in the list V with the values in the set S.
evalV1,S = 417−317
evalV2,S = 317217
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