Dados os vetores u = (1,2,3), v = (3,2,0) e w = (2,0,0) determine números α, β, γ tais que αu + βv + γw = (1,1,1).

-> αu + βv + γw = (1,1,1)

-> α(1,2,3) + β(3,2,0) + γ(2,0,0) = (1,1,1)

-> (α,2α,3α) + (3β,2β,0) + (2γ,0,0) = (1,1,1)

-> (α+3β+2γ, 2α+2β, 3α) = (1,1,1)

Valor de α:

-> 3α = 1

-> α = 1/3

Valor de β:

-> 2α + 2β = 1

-> α + β = 1/2

-> β = 1/2 - α

-> β = 1/2 - 1/3

-> β = 1/6

Valor de γ:

-> α + 3β + 2γ = 1

-> 2γ = 1 - α - 3β

-> 2γ = 1 - 1/3 - 3*1/6

-> 2γ = 1 - 1/3 - 1/2

-> 2γ = 1 - 5/6

-> 2γ = 1/6

-> γ = 1/12

Solução:

{ α = 1/3

{ β = 1/6

{ γ = 1/12

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