> [!NOTE] Definition > Let $n$ be a positive [[Natural Numbers|natural number]]. The set $\mathbb{R}^{n},$ which is the set of all $n$-[[List|tuples]] (or vectors of length $n$) of [[Real numbers|real numbers]], given by $\mathbb{R}^{n}=\left\{\left.\left(\begin{array}{c}a_1 \\ a_2 \\ \vdots \\a_{n} \end{array}\right) \right\rvert\, \forall i=1, \ldots, n: a_i \in \mathbb{R} \right\}$is called the real $n$-space (or real Cartesian space). The positive integer $n$ is known as its **dimension.** # Properties We define [[Addition in Real n-Space]] by ... and [[Scalar Multiplication in Real n-Space]] by ... By [[Real n-Space is Finite Dimensional Real Vector Space]], ... By [[Scalar Multiplication by Zero in Real n-Space]], for all $\underline{v}\in\mathbb{R}^{n},$ $0\underline{v}=\underline{0}$ where $\underline{0}=(0,0,\dots,0).$ By [[Real n-Space is Euclidean Space]], $\mathbb{R}^{n}$ together with [[Dot Product in Real n-Space]] forms a [[Euclidean spaces]].