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📰 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) $.
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