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

["Finding Vector $ \overrightarrow{OC} = m\mathbf{a} + n\mathbf{b} $ Perpendicular to $ \mathbf{a} - \mathbf{b} $: A Solution with Key Constraints", "In vector geometry, determining a specific vector lying on a plane often hinges on the condition that it is perpendicular to a given direction. Consider triangle points $ A $ and $ B $ defined by position vectors $ \vec{OA} = \mathbf{a} $ and $ \vec{OB} = \mathbf{b} $. Let $ C $ be a point such that $ \overrightarrow{OC} = m\mathbf{a} + n\mathbf{b} $, and it is required that $ \overrightarrow{OC} \perp (\mathbf{a} - \mathbf{b}) $.", "### The Perpendicularity Condition", "Perpendicular vectors have a dot product of zero. Thus,\n$$\n\overrightarrow{OC} \cdot (\mathbf{a} - \mathbf{b}) = 0\n$$\nSubstituting $ \overrightarrow{OC} = m\mathbf{a} + n\mathbf{b} $, we get:\n$$\n(m\mathbf{a} + n\mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = 0\n$$\nExpanding the dot product using distributive property:\n$$\nm\mathbf{a} \cdot \mathbf{a} - m\mathbf{a} \cdot \mathbf{b} + n\mathbf{b} \cdot \mathbf{a} - n\mathbf{b} \cdot \mathbf{b} = 0\n$$\nThis simplifies to:\n$$\nm|\mathbf{a}|^2 - m(\mathbf{a} \cdot \mathbf{b}) + n(\mathbf{b} \cdot \mathbf{a}) - n|\mathbf{b}|^2 = 0\n$$", "### Substituting Known Values", "Given:\n- $ |\mathbf{a}| = 2 \Rightarrow |\mathbf{a}|^2 = 4 $\n- $ |\mathbf{b}| = 3 \Rightarrow |\mathbf{b}|^2 = 9 $\n- $ \mathbf{a} \cdot \mathbf{b} = 3 $ (since $ \cos 60^\circ = \frac{1}{2} $, $ 2 \cdot 3 \cdot \frac{1}{2} = 3 $)", "Plug these into the equation:\n$$\n4m - 3m + 3n - 9n = 0\n$$\n$$\n(4m - 3m) + (3n - 9n) = 0 \Rightarrow m - 6n = 0\n$$\nThus, $ m = 6n $.", "Choosing $ n = 1 $ for simplicity, we find $ m = 6 $. Therefore, the vector is:\n$$\n\overrightarrow{OC} = 6\mathbf{a} + 1\mathbf{b}\n$$", "This relationship $ m = 6n $ governs all such vectors — any scalar multiple of $ (6, 1) $ satisfies the perpendicularity condition. Setting $ n = 1 $, $ m = 6 $ provides a clear representative solution.", "### Final Notes", "While further normalization of $ \overrightarrow{OC} $ may be possible depending on scale requirements, the core constraint $ m = 6n $ remains invariant. This foundational relation enables flexible construction of valid points $ C $ in the plane defined by $ \mathbf{a} $ and $ \mathbf{b} $, essential in geometric modeling and vector analysis.", "Key takeaway: When determining a vector perpendicular to $ \mathbf{a} - \mathbf{b} $ expressed as $ m\mathbf{a} + n\mathbf{b} $, the perpendicularity condition yields a linear equation between $ m $ and $ n $. Solving it yields a powerful parametric solution — $ m = 6n $ — simplifying vector construction in geometric problems."]