设向量a=(cosα,sinα),b=(cosβ,sinβ)(1)若a-b=(-2/3,1/3),求cos(2)若cos=60°,那么t为何值│a-tb│的值最小?
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![设向量a=(cosα,sinα),b=(cosβ,sinβ)(1)若a-b=(-2/3,1/3),求cos(2)若cos=60°,那么t为何值│a-tb│的值最小?](/uploads/image/z/4033451-11-1.jpg?t=%E8%AE%BE%E5%90%91%E9%87%8Fa%3D%28cos%CE%B1%2Csin%CE%B1%29%2Cb%3D%28cos%CE%B2%2Csin%CE%B2%29%EF%BC%881%EF%BC%89%E8%8B%A5a-b%3D%28-2%2F3%2C1%2F3%29%2C%E6%B1%82cos%EF%BC%882%EF%BC%89%E8%8B%A5cos%3D60%C2%B0%2C%E9%82%A3%E4%B9%88t%E4%B8%BA%E4%BD%95%E5%80%BC%E2%94%82a-tb%E2%94%82%E7%9A%84%E5%80%BC%E6%9C%80%E5%B0%8F%3F)
设向量a=(cosα,sinα),b=(cosβ,sinβ)(1)若a-b=(-2/3,1/3),求cos(2)若cos=60°,那么t为何值│a-tb│的值最小?
设向量a=(cosα,sinα),b=(cosβ,sinβ)
(1)若a-b=(-2/3,1/3),求cos
(2)若cos=60°,那么t为何值│a-tb│的值最小?
设向量a=(cosα,sinα),b=(cosβ,sinβ)(1)若a-b=(-2/3,1/3),求cos(2)若cos=60°,那么t为何值│a-tb│的值最小?
(1) cos = a•b /∣a∣∣b∣
= (cosα,sinα)•(cosβ,sinβ) / [√(cos²α+sin²α) * √(cos²β+sin²β)]
= (cosαcosβ + sinαsinβ) / √1 * √1
= cosαcosβ + sinαsinβ
∵a - b = (-2/3,1/3)
∴(cosα,sinα) - (cosβ,sinβ) = (-2/3,1/3)
(cosα-cosβ,sinα-sinβ) = (-2/3,1/3)
比较系数,得cosα - cosβ = -2/3 (1)
sinα - sinβ = 1/3 (2)
(1)² + (2)²:(cos²α - 2cosαcosβ + cos²β) + (sin²α - 2sinαsinβ + sin²β) = 4/9 + 1/9
化简:2 - 2(cosαcosβ + sinαsinβ) = 5/9
得:cosαcosβ + sinαsinβ = 13/18
∴所求:cos = 13/18
(2) ∣a - tb∣= √(a - tb)²
= √(a² - 2ta•b + b²)
= √(∣a∣² - 2t∣a∣∣b∣cos + ∣b∣²)
= √[(cos²α+sin²α) - 2t√(cos²α+sin²α)*√(cos²β+sin²β)cos60° + (cos²β+sin²β)]
= √(1 - 2t * 1 * 1 * 1/2 + 1)
= √(2 - t)
∵∣a - tb∣≥ 0
∴ √(2 - t)≥ 0
得t = 2时,∣a - tb∣取得最小值.