> [!Definition] (Dot Product) > Let $n \geq 1$ and $\underline{v}= (a_{1},\dots,a_{n})^{T}$, $\underline{w}=(b_{1},\dots,b_{n})^{T} \in \mathbb{R}^{n}$ be elements of the [[Real n-Space|real n-space]]. The *dot product* of $\underline{v}$ and $\underline{w}$, denoted $\underline{v} \cdot \underline{w}$ is the scalar $\underline{v} \cdot \underline{w} = \sum_{i=1}^{m} a_{i}b_{i} \in \mathbb{R}.$ # Properties By [[Dot Product is an Inner Product on Real n-Space]], dot product is commutative (for all $\underline{v},\underline{w}\in \mathbb{R}^{n},$ $\underline{v}\cdot \underline{w}=\underline{w}\cdot \underline{v}$), non-negative definite (for all $\underline{v}\in \mathbb{R}^{n},$ $\underline{v}\cdot \underline{v}\geq 0$) and bilinear (for all $\lambda_{1},\lambda_{2}\in\mathbb{R},$ $\underline{v}_{1},\underline{v}_{2},\underline{w}\in \mathbb{R}^{n},$ $(\lambda v_{1}+\lambda v_{2})\cdot w=\lambda_{1}(v_{1}\cdot \underline{w})+\lambda_{2}(\underline{v}_{2}\cdot \underline{w})$ ).