A weighted average is a specialized form of a weighted sum where all weights are non-negative and sum to exactly $$1$$ (i.e., $$\sum_{i=1}^{n} w_i = 1$$). When a weight vector $$\mathbf{w}$$ meets these criteria, the dot product $$\mathbf{x}^	op \mathbf{w}$$ with a value vector $$\mathbf{x}$$ computes the weighted average of those values.

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A fundamental application of the dot product is the computation of a weighted sum. Given a vector of values $$\mathbf{x} \in \mathbb{R}^n$$ and a corresponding vector of weights $$\mathbf{w} \in \mathbb{R}^n$$, the weighted sum of the elements in $$\mathbf{x}$$ based on the weights in $$\mathbf{w}$$ can be concisely represented as the dot product $$\mathbf{x}^	op \mathbf{w}$$.

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