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Question Number 196440 by mokys last updated on 24/Aug/23

prove ∣v−w∣≥∣v∣−∣w∣

prove∣v−w∣⩾∣v∣−∣w∣

Answered by Frix last updated on 24/Aug/23

v, w ∈C v=ae^(iα) ∧w=be^(iβ) ∧a, b ≥0 ∣v−w∣≥∣v∣−∣w∣ (√(a^2 −2abcos (α−β) +b^2 ))≥a−b Case 1: a−b<0 True because (√(...))≥0 C.ase 2: a−b≥0 a^2 −2abcos (α−β) +b^2 ≥a^2 −2ab−b^2 −2abcos (α−β) ≥−2ab cos (α−β) ≤1 true

v,w∈Cv=aeiα∧w=beiβ∧a,b⩾0∣v−w∣⩾∣v∣−∣w∣a2−2abcos(α−β)+b2⩾a−bCase1:a−b<0Truebecause...⩾0C.ase2:a−b⩾0a2−2abcos(α−β)+b2⩾a2−2ab−b2−2abcos(α−β)⩾−2abcos(α−β)⩽1true

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