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) $.

["Optimize Crop Growth Vectors: Solving for $ m $ and $ n $ in a Biophysical Model", "Understanding plant architecture is essential in agricultural biology, especially when studying how crop growth vectors influence light capture and yield. In a recent model, a plant biologist analyzes the growth direction vector $ \vec{OG} = m\vec{OA} + n\vec{OB} $, where $ \vec{OA} $ and $ \vec{OB} $ are perpendicular vectors representing directional growth components, with magnitudes $ |\vec{OA}| = 3 $, $ |\vec{OB}| = 4 $, and $ \vec{OA} \cdot \vec{OB} = 0 $. Given $ |\vec{OG}| = 5 $ and $ \vec{OG} \perp (\vec{OA} + \vec{OB}) $, determine the constants $ m $ and $ n $.", "### Step 1: Use Orthogonality Condition", "Since $ \vec{OG} $ is perpendicular to $ \vec{OA} + \vec{OB} $, their dot product is zero:", "$$\n\vec{OG} \cdot (\vec{OA} + \vec{OB}) = 0\n$$", "Substitute $ \vec{OG} = m\vec{OA} + n\vec{OB} $:", "$$\n(m\vec{OA} + n\vec{OB}) \cdot (\vec{OA} + \vec{OB}) = 0\n$$", "Expand using distributivity and dot product linearity:", "$$\nm\vec{OA} \cdot \vec{OA} + n\vec{OB} \cdot \vec{OA} + m\vec{OA} \cdot \vec{OB} + n\vec{OB} \cdot \vec{OB} = 0\n$$", "Given $ \vec{OA} \cdot \vec{OB} = 0 $, all cross terms vanish:", "$$\nm|\vec{OA}|^2 + n|\vec{OB}|^2 = 0\n$$", "Substitute $ |\vec{OA}|^2 = 9 $, $ |\vec{OB}|^2 = 16 $:", "$$\n9m + 16n = 0 \quad \ ext{(Equation 1)}\n$$", "### Step 2: Use the Magnitude Condition", "Given $ |\vec{OG}| = 5 $, so $ |\vec{OG}|^2 = 25 $. Compute:", "$$\n|\vec{OG}|^2 = (m\vec{OA} + n\vec{OB}) \cdot (m\vec{OA} + n\vec{OB}) = m^2|\vec{OA}|^2 + n^2|\vec{OB}|^2\n$$", "Since $ \vec{OA} \perp \vec{OB} $:", "$$\n|\vec{OG}|^2 = 9m^2 + 16n^2 = 25 \quad \ ext{(Equation 2)}\n$$", "### Step 3: Solve the System of Equations", "From Equation 1:\n$$\nm = -\frac{16}{9}n\n$$", "Substitute into Equation 2:", "$$\n9\left(-\frac{16}{9}n\right)^2 + 16n^2 = 25\n\Rightarrow 9 \cdot \frac{256}{81}n^2 + 16n^2 = 25\n\Rightarrow \frac{2304}{81}n^2 + 16n^2 = 25\n$$", "Convert $ 16 $ to $ \frac{1296}{81} $:", "$$\n\left( \frac{2304 + 1296}{81} \right)n^2 = 25\n\Rightarrow \frac{3600}{81}n^2 = 25\n\Rightarrow n^2 = 25 \cdot \frac{81}{3600} = \frac{2025}{3600} = \frac{81}{144} = \frac{9}{16}\n$$", "So $ n = \pm \frac{3}{4} $", "Now find $ m $:", "If $ n = \frac{3}{4} $, then $ m = -\frac{16}{9} \cdot \frac{3}{4} = -\frac{48}{36} = -\frac{4}{3} $", "If $ n = -\frac{3}{4} $, then $ m = -\frac{16}{9} \cdot (-\frac{3}{4}) = \frac{48}{36} = \frac{4}{3} $", "Both satisfy $ 9m + 16n = 0 $. But since $ \vec{OA} $ and $ \vec{OB} $ represent growth directions and $ \vec{OG} $ is a linear combination representing net growth, both sign combinations are mathematically valid. However, in biological context, positive coefficients often reflect forward growth—thus the positive direction alignment with $ \vec{OB} $ when $ \vec{OA} $ is orthogonal supports $ n > 0 $. Both solutions are valid geometrically, but standard convention chooses the solution with consistent orientation.", "Check magnitude for $ \left(m, n\right) = \left(-\frac{4}{3}, \frac{3}{4}\right) $:", "$$\n\vec{OG}^2 = 9\left(\frac{16}{9}\right) + 16\left(\frac{9}{16}\right) = 16 + 9 = 25 \quad \ ext{✓}\n$$", "Similarly for $ \left(\frac{4}{3}, -\frac{3}{4}\right) $: same result.", "But since the perpendicularity and magnitude are symmetric, both yield valid solutions. However, standard vector decomposition prefers same sign alignment unless constrained. Here, no contradiction, so both mathematically correct. but typically in bloom orientation models, positive coupling with $ \vec{OB} $ is expected—still, we report the solution satisfying the system.", "But wait: from $ 9m + 16n = 0 $, $ m = -\frac{16}{9}n $, plug into magnitude:", "We already solved: gives $ n = \pm \frac{3}{4} $, $ m = \mp \frac{4}{3} $", "Both valid. However, in context, if $ \vec{OA} $ and $ \vec{OB} $ define orthogonal growth axes (say north and east), and $ \vec{OG} $ is perpendicular to northeast, then $ \vec{OG} $ should have components in same signs as $ \vec{OB} $ if in south-east quadrant—but no direct sign constraint.", "But since magnitude combines via squares, both satisfy. But let’s compute the vector direction.", "For $ m = -\frac{4}{3}, n = \frac{3}{4} $: $ \vec{OG} $ points southwest, opposite $ \vec{OA} $, but magnitude: $ \sqrt{9 \cdot \frac{16}{9} + 16 \cdot \frac{9}{16}} = \sqrt{16 + 9} = 5 $ — valid.", "Same for $ m = \frac{4}{3}, n = -\frac{3}{4} $: $ \vec{OG} $ points northwest.", "No biological exclusion. But in standard vector decomposition in plant architecture, coefficients reflect real directed growth; without directional constraint, both are mathematically acceptable. However, the problem asks for the solution — implying uniqueness.", "But observe: $ \vec{OG} = m\vec{OA} + n\vec{OB} $, with $ \vec{OA} \perp \vec{OB} $, and $ |\vec{OA}| = 3 $, $ |\vec{OB}| = 4 $. This is a right triangle setup.", "The vector $ \vec{OG} $ has magnitude 5 — like a 3-4-5 triangle. Indeed $ 3^2 + 4^2 = 5^2 $. So"]