Provide two vectors. We compute ⟨x,y⟩, norms, explain why the inequality holds, and run checks.
Answer
⟨x,y⟩
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‖x‖
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‖y‖
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‖x‖·‖y‖
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Slack (bound − |⟨x,y⟩|)
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Verdict
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Reason Why
For any real t, ‖x − t y‖² ≥ 0 expands to ‖x‖² − 2t⟨x,y⟩ + t²‖y‖² ≥ 0. The discriminant is ≤ 0, hence
(2⟨x,y⟩)² ≤ 4‖x‖²‖y‖², i.e. |⟨x,y⟩| ≤ ‖x‖‖y‖. Equality holds iff x and y are linearly dependent (x = λy).