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Understanding vector addition is crucial as it forms the base for many operations involving vectors, including the dot product. In simple terms, adding two or more vectors together means summing their corresponding components. For instance, if you have two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), their sum \( \mathbf{a} + \mathbf{b} \) is computed as \( \langle a_1+b_1, a_2+b_2, a_3+b_3 \rangle \).
This operation extends naturally to vectors in higher dimensions, such as \( n \)-dimensional space. When performing vector addition, each component is treated independently, making the operation straightforward and intuitive.
The resulting vector from adding two vectors represents a new magnitude and direction, effectively combining the effects of the original vectors. In graphical terms, this can be visualized as placing the tail of one vector at the head of the other, resulting in a diagonal when completed to a parallelogram.

The distributive property in the context of vectors and dot products is analogous to distribution in algebra. Specifically, when dealing with dot products, this property allows a vector to be distributed across a sum, yielding the same result as the sum of individual dot products. Mathematically, this is illustrated with \( \mathbf{x} \cdot (\mathbf{y} + \mathbf{z}) = (\mathbf{x} \cdot \mathbf{y}) + (\mathbf{x} \cdot \mathbf{z}) \).
This means you can compute a dot product over a sum by separately computing dot products and then adding them together. This property is a crucial simplifier in vector calculus, making computations easier by breaking them into smaller pieces.
It's important to note that this property works whether the distribution is from the right or the left, as seen in the original exercise, where both \( (\mathbf{u}+\mathbf{v}) \cdot \mathbf{w} \) and \( \mathbf{u} \cdot (\mathbf{v}+\mathbf{w}) \) return the same result, ensuring consistency in multivariable calculus.

Multivariable calculus extends the principles of single-variable calculus into higher dimensions. Vectors play a significant role in this area as they allow for the manipulation and analysis of multi-dimensional data. In addition to handling vectors, multivariable calculus explores functions with multiple inputs and outputs.
Such analysis often involves computing with vectors, where operations like the dot product become essential. The dot product, being an operation that produces a scalar from two vectors, is heavily used in determining angles, projections, and lengths in multi-dimensional space. This is vital when dealing with gradient vectors, divergence, and curl, all of which require a solid understanding of vector operations.
Moreover, the dot product's distributive property fits seamlessly into multivariable calculus. It simplifies the calculus of vector functions, makes it easier to handle linear transformations, and aids in the application of theorems like Green's and Stokes'. Understanding these connections can help students appreciate the power and utility of calculus in analyzing complex, real-world systems.