# Definition(s) > [!NOTE] Definition 1 (Line in $\mathbb{R}^n$) > A line in $\mathbb{R}^n$ is a subset of $\mathbb{R}^m$ of the form $\{ \underline{u}+\lambda \underline{v} \mid \lambda \in \mathbb{R}^n \}$ for some $\underline{u},\underline{v}\in \mathbb{R}^n.$ > > $\underline{v}$ is known as its direction vector. > [!NOTE] Definition 2 (Line in $\mathbb{R}^n$) > The line in $\mathbb{R}^n$ through distinct $\mathbf{x},\mathbf{y}$ is given by $\{ \mathbf{z} \in \mathbb{R}^n \mid d(\mathbf{x},\mathbf{y}) = d(\mathbf{x},\mathbf{z})+d(\mathbf{z},\mathbf{y}) \}$ > > where $d$ denotes the [[Euclidean Metric on Real n-Space|standard Euclidean metric]] on $\mathbb{R}^n.$ > > # Properties(s) # Application(s) **More examples**: # Bibliography