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\mathbf{u} \cdot \mathbf{u} = 4,\quad \|\mathbf{v}\|^2 = 9,\quad \|\mathbf{w}\|^2 = 16,
\mathbf{u} \cdot \mathbf{v} = 1,\quad \mathbf{v} \cdot \mathbf{w} = 6.
Assume all vectors lie in a 3D subspace. Express \(\mathbf{u}\) and \(\mathbf{w}\) in terms of an orthonormal basis adapted to the constraints. Use geometric insight: the maximum \(\mathbf{u} \cdot \mathbf{w}\) occurs when \(\mathbf{u}\) and \(\mathbf{w}\) are aligned as much as possible, consistent with intermediate alignments to \(\mathbf{v}\).

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