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📰 Question: In a diagram, $ \|\overrightarrow{OA}\| = 2 $, $ \|\overrightarrow{OB}\| = 3 $, and $ \angle AOB = 60^\circ $. If $ \overrightarrow{OC} = m\overrightarrow{OA} + n\overrightarrow{OB} $, find $ (m, n) $ such that $ \overrightarrow{OC} $ is perpendicular to $ \overrightarrow{OA} - \overrightarrow{OB} $, modeling directional balance in ecological data. 📰 Solution: Let $ \vec{OA} = \mathbf{a} $, $ \vec{OB} = \mathbf{b} $. Then $ \overrightarrow{OC} = m\mathbf{a} + n\mathbf{b} $. For $ \overrightarrow{OC} \perp (\mathbf{a} - \mathbf{b}) $, their dot product is zero: $ (m\mathbf{a} + n\mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = 0 $. Expand: $ m\|\mathbf{a}\|^2 - m\mathbf{a} \cdot \mathbf{b} + n\mathbf{b} \cdot \mathbf{a} - n\|\mathbf{b}\|^2 = 0 $. Substitute $ \|\mathbf{a}\| = 2 $, $ \|\mathbf{b}\| = 3 $, $ \mathbf{a} \cdot \mathbf{b} = 2 \cdot 3 \cdot \cos 60^\circ = 3 $: $ 4m - 3m + 3n - 9n = 0 \Rightarrow m - 6n = 0 $. Choose $ n = 1 $, then $ m = 6 $. Normalize if needed, but the relation is $ m = 6n $. For simplicity, take $ n = 1 $, $ m = 6 $. 📰 \boxed{(6, 1)}Question: A hydrologist studying groundwater flow measures the volume of water (in liters) passing through a sensor at different times, recording values that are positive multiples of 5. If the cube of one such measurement is less than 1500, what is the greatest possible value of that measurement? 📰 How To Create Templates In Outlook 📰 Regex For Excel 2760424 📰 You Wont Believe How Fidelity Hra Unlocks Life Changing Financial Perks 3595912 📰 Shock Moment Free Fortnite Crosshairs And Experts Are Concerned 📰 Chainlink Stock 6709434 📰 Apps To Transfer Cash 📰 Steam Resident Evil Hd Remaster 📰 Solo Leveling Release Time 📰 Are Steam Games Free 📰 Child First 📰 Rec Fuel Near Me 329782 📰 What Month Is 10 📰 Log In Verizon Account 📰 Good Health Insurance In Florida 📰 Great Games From Ps2