a b是两个不共线的单位向量,向量c满足

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|c|²=λ²+(1-λ)²+2λ(1-λ)(a.b)=1/4
得 2(a.b) = [1/4 - λ² - (1-λ)²]/[λ(1-λ)]
|a-b|²=1+1-2(a.b)=2-2(a.b) = 2 - [1/4 - λ² - (1-λ)²]/[λ(1-λ)]
= 0.75/[λ(1-λ)]
= 0.75/[0.25-(λ-0.5)²]
|a-b|最小,则必须满足 λ=1/2
此时 2(a.b) = [1/4 - λ² - (1-λ)²]/[λ(1-λ)] = -1
(a.b) = -1/2 ,即cos = -1/2
= ±120° + 360°k (k∈Z)
考虑到向量a,b夹角都在0~180°之间,故 = 120°