Solution: The dot product \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\). Since \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are unit vectors, each dot product is at most 1. The maximum occurs when \(\mathbf{u}\) aligns with both \(\mathbf{v}\) and \(\mathbf{w}\), i.e., \(\mathbf{v} = \mathbf{w} = \mathbf{u}\). Then \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = 1 + 1 = 2\).

["Understanding the Dot Product Distributive Property and Its Maximum Value", "In vector mathematics, one of the most powerful and intuitive properties is the distributive nature of the dot product. For unit vectors (\mathbf{u}), (\mathbf{v}), and (\mathbf{w}), the identity\n[\n\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\n]\nplays a fundamental role in simplifying expressions and optimizing vector-based computations. This article explores this vector identity, explains why each dot product is bounded above by 1, and determines the maximum possible value of the expression when (\mathbf{u}), (\mathbf{v}), and (\mathbf{w}) are unit vectors.", "---", "### The Dot Product and Unit Vectors", "The dot product of two vectors (\mathbf{a} \cdot \mathbf{b}) measures the projection of one vector onto another, scaled by their magnitudes. Mathematically,\n[\n\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos\ heta,\n]\nwhere (\ heta) is the angle between them. For unit vectors—that is, vectors with magnitude 1—the formula simplifies to\n[\n\mathbf{a} \cdot \mathbf{b} = \cos\ heta.\n]\nSince cosine values range between (-1) and (1), each dot product achieves a maximum value of 1 (when vectors are parallel and point in the same direction) and a minimum of (-1) (when opposite).", "---", "### Applying the Distributive Property", "Given that (\mathbf{u}), (\mathbf{v}), and (\mathbf{w}) are unit vectors, the distributive law allows us to expand:\n[\n\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}.\n]\nThis expresses the projection of (\mathbf{u}) onto the sum of two vectors as the sum of its projections onto each vector individually.", "---", "### Maximizing the Expression", "To find the maximum value of (\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})), we consider the constraints imposed by unit vectors. Since each dot product cannot exceed 1 in absolute value,\n[\n\mathbf{u} \cdot \mathbf{v} \leq 1 \quad \ ext{and} \quad \mathbf{u} \cdot \mathbf{w} \leq 1.\n]\nThus, the upper bound of the sum is\n[\n\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) \leq 1 + 1 = 2.\n]", "This maximum is attainable only when both (\mathbf{u} \cdot \mathbf{v} = 1) and (\mathbf{u} \cdot \mathbf{w} = 1). Recall that (\mathbf{u} \cdot \mathbf{v} = 1) when (\mathbf{u} = \mathbf{v}), and similarly (\mathbf{u} \cdot \mathbf{w} = 1) when (\mathbf{u} = \mathbf{w}). Therefore, the maximum value of 2 occurs precisely when\n[\n\mathbf{v} = \mathbf{w} = \mathbf{u}.\n]", "Substituting,\n[\n\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot (\mathbf{u} + \mathbf{u}) = \mathbf{u} \cdot (2\mathbf{u}) = 2(\mathbf{u} \cdot \mathbf{u}) = 2 \ imes 1 = 2.\n]", "---", "### Practical Implications", "This identity and its maximum demonstrate how vector alignment can simplify and optimize calculations in physics, engineering, computer graphics, and machine learning—fields heavily relying on direction and magnitude. Recognizing when vectors align enables precise control over projections and energy computations, such as work done ((\mathbf{F} \cdot \mathbf{d})) or signal processing with inner products.", "---", "### Summary", "- The distributive property (\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}) holds for unit vectors.\n- Each dot product achieves a maximum of 1.\n- The sum reaches maximum value of 2 when (\mathbf{v}) and (\mathbf{w}) both align perfectly with (\mathbf{u}), i.e., (\mathbf{v} = \mathbf{w} = \mathbf{u}).\n- Understanding this property improves problem-solving involving vector projections and optimizations.", "---", "### Key Takeaway", "For unit vectors (\mathbf{u}), (\mathbf{v}), and (\mathbf{w}), the expression (\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})) achieves its maximum value of 2 precisely when (\mathbf{v}) and (\mathbf{w}) point in the same direction as (\mathbf{u}), illustrating the power and elegance of the dot product’s distributive nature.", "Keywords: dot product, unit vectors, distributive property, maximum value, vector algebra, vector projection, inner product."]