> [!NOTE] Definition (Length) > We define the *length* $|| \;||:\mathbb{R}^{n}\to \mathbb{R}$ of a [[Vector|vector]] $\underline{v}\in \mathbb{R}^{n}$, denoted $\lvert \rvert v \lvert \rvert$ (aka *norm* of $\underline{v}$), to be $\lvert \rvert v \lvert \rvert = \sqrt{ \underline{v} \cdot \underline{v} } $where $\underline{v}\cdot \underline{v}$ denotes a [[Dot Product in Real n-Space|dot product]]. # Properties By [[Non-negative Definiteness of Length of Real Vector]], for all $\underline{v}\in \mathbb{R}^{n},$ $||\underline{v}||\geq 0$ with equality iff $\underline{v}=\underline{0}.$ By [[Length of Scaled Real Vector]], for all $\lambda\in \mathbb{R},$ $\underline{v}\in \mathbb{R}^{n},$ $||\lambda \underline{v}||=|\lambda|||\underline{v}||.$