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Home » Dance Like the Wind: The Dancer of the Boreal Valley Becomes an Unstoppable Icon - go-checkin.com

Dance Like the Wind: The Dancer of the Boreal Valley Becomes an Unstoppable Icon - go-checkin.com

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \boxed{540^\circ}

📌 Question: A plant biologist studies the growth vector $ \vec{OG} $ of a crop, defined by $ \vec{OG} = m\vec{OA} + n\vec{OB} $, where $ \|\vec{OA}\| = 3 $, $ \|\vec{OB}\| = 4 $, and $ \vec{OA} \cdot \vec{OB} = 0 $. If $ \|\vec{OG}\| = 5 $ and $ \vec{OG} $ is perpendicular to $ \vec{OA} + \vec{OB} $, find $ (m, n) $.

📌 Solution: Since $ \vec{OA} \cdot \vec{OB} = 0 $, they are perpendicular. Let $ \vec{OA} = (3, 0) $, $ \vec{OB} = (0, 4) $. Then $ \vec{OG} = (3m, 4n) $.

📌 Perpendicularity condition: $ \vec{OG} \cdot (\vec{OA} + \vec{OB}) = 0 \implies (3m, 4n) \cdot (3, 4) = 9m + 16n = 0 $.

📌 Magnitude condition: $ \sqrt{(3m)^2 + (4n)^2} = 5 \implies 9m^2 + 16n^2 = 25 $.

📌 From $ 9m + 16n = 0 $, solve $ m = -\frac{16}{9}n $. Substitute into the magnitude equation:

📌 9\left(\frac{256}{81}n^2\right) + 16n^2 = \frac{256}{9}n^2 + 16n^2 = \frac{256 + 144}{9}n^2 = \frac{400}{9}n^2 = 25.

📌 Thus $ n^2 = \frac{25 \cdot 9}{400} = \frac{225}{400} = \frac{9}{16} \implies n = \pm \frac{3}{4} $.

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