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student18401

Geometric proofs

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  • leung_po_chuen
    2020-08-25 15:05:20

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  • 解數雜貨店
    2020-08-26 08:44:26
    a) βPM = αMQ β(OP-OM) = α(OM-OQ) β(p-m) = α(m-q) bi) let v be the position vector Decompose v into two vectors, v = λ₁a + λ₂b angle between v and a is the same as the angle between v and b, v⋅a = v⋅b (λ₁a+λ₂b)⋅a = (λ₁a+λ₂b)⋅b λ₁(a⋅a)+λ₂(a⋅b) = λ₁(a⋅b)+λ₂(b⋅b) (λ₁-λ₂)+(λ₂-λ₁)(a⋅b) = 0 (λ₁-λ₂)(1-a⋅b) = 0 a⋅b ≠ 1 because the angle between a and b is not 0. ∴ λ₁-λ₂ = 0 λ₁ = λ₂ v can be written as λ(a + b) bii) m = (βp+αq)/(α+β) = (βk/(α+β))a + (αℓ/(α+β))b From bi, we know that βk/(α+β) = αℓ/(α+β) βk = αℓ α/β = k/ℓ
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