Euclidean SpaceDefinition 1. $\forall n \in \mathbb{N}$, let $$\mathbb{R}^n := \{ \textbf{x} \,|\, \textbf{x} = (x_1, ..., x_n) \, \text{where} \, x_i \in \mathbb{R} (i = 1, 2, ..., n)\}.$$ For $\textbf{x} = (x_1, ..., x_n)$ and $\textbf{y} = (y_1, ..., y_n) \in\mathbb{R}^n$, we define the coordinatewise operations: $$\textbf{x} + \textbf{y} = (x_1 + y_1, ..., x_n + y_n) \\ \alpha \textbf{x} = (..