Determine:
(1) (1,2,3) wed>=(3,1,2)
(2) (1,2,3) wed>=(2,4,6)
(3) (3,1,2) wed>=(2,4,6)
(4) (-1,1,2) wed>=(-1,1,2)
(5) (0,4,2) wed>=(1,3,2)
(6) (1,2,5) wed>=(2,0,-1)
(7) (6,-2,-4) wed>=(-1,-2,1)
(8) (7,0,-5) wed>=(1,2,-1)
(9) (1,-3,1) wed>=(1,1,4)

Passo 1

O enunciado pede para calcular o produto exterior (wedge) de pares de vetores em R^3. O conteúdo é álgebra exterior, e vamos usar a definição que associa a cada par de vetores um bivetor com componentes nas bases e1wed>=e2, e1wed>=e3 e e2wed>=e3.

Passo 2

Em R^3, dados
v=(v1,v2,v3), w=(w1,w2,w3),
o produto exterior é
vwed>=w =(v1w2 - v2w1),e1wed>=e2 +(v1w3 - v3w1),e1wed>=e3 +(v2w3 - v3w2),e2wed>=e3

Passo 3

Aplicando essa fórmula a cada par:
(1) ,(1,2,3)wed>=(3,1,2):
1cdot1 - 2cdot3 = -5, 1cdot2 - 3cdot3 = -7, 2cdot2 - 3cdot1 = 1
(2) ,(1,2,3)wed>=(2,4,6):
1cdot4 - 2cdot2 = 0, 1cdot6 - 3cdot2 = 0, 2cdot6 - 3cdot4 = 0
(3) ,(3,1,2)wed>=(2,4,6):
3cdot4 - 1cdot2 = 10, 3cdot6 - 2cdot2 = 14, 1cdot6 - 2cdot4 = -2
(4) ,(-1,1,2)wed>=(-1,1,2):
(-1)cdot1 - 1cdot(-1)=0, (-1)cdot2 - 2cdot(-1)=0, 1cdot2 - 2cdot1=0
(5) ,(0,4,2)wed>=(1,3,2):
0cdot3 - 4cdot1 = -4, 0cdot2 - 2cdot1 = -2, 4cdot2 - 2cdot3 = 2
(6) ,(1,2,5)wed>=(2,0,-1):
1cdot0 - 2cdot2 = -4, 1cdot(-1) - 5cdot2 = -11, 2cdot(-1) - 5cdot0 = -2
(7) ,(6,-2,-4)wed>=(-1,-2,1):
6cdot(-2) - (-2)cdot(-1) = -14, 6cdot1 - (-4)cdot(-1) = 2, (-2)cdot1 - (-4)cdot(-2) = -10
(8) ,(7,0,-5)wed>=(1,2,-1):
7cdot2 - 0cdot1 = 14, 7cdot(-1) - (-5)cdot1 = -2, 0cdot(-1) - (-5)cdot2 = 10
(9) ,(1,-3,1)wed>=(1,1,4):
1cdot1 - (-3)cdot1 = 4, 1cdot4 - 1cdot1 = 3, (-3)cdot4 - 1cdot1 = -13

Resposta

(1)
(1,2,3)wed>=(3,1,2) =-5,e1wed>=e2 -7,e1wed>=e3 +1,e2wed>=e3
(2)
(1,2,3)wed>=(2,4,6)=0
(3)
(3,1,2)wed>=(2,4,6) =10,e1wed>=e2 +14,e1wed>=e3 -2,e2wed>=e3
(4)
(-1,1,2)wed>=(-1,1,2)=0
(5)
(0,4,2)wed>=(1,3,2) =-4,e1wed>=e2 -2,e1wed>=e3 +2,e2wed>=e3
(6)
(1,2,5)wed>=(2,0,-1) =-4,e1wed>=e2 -11,e1wed>=e3 -2,e2wed>=e3
(7)
(6,-2,-4)wed>=(-1,-2,1) =-14,e1wed>=e2 +2,e1wed>=e3 -10,e2wed>=e3
(8)
(7,0,-5)wed>=(1,2,-1) =14,e1wed>=e2 -2,e1wed>=e3 +10,e2wed>=e3
(9)
(1,-3,1)wed>=(1,1,4) =4,e1wed>=e2 +3,e1wed>=e3 -13,e2wed>=e3

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