> [!NOTE] Definition (Euclidean Vector) > Euclidean vector is a [[Vectors|vector]] over the reals (a member of $\mathbb{R}^{n}$). > [!Notation] Notation & Language > - If write the elements of $\mathbb{R}^{n}$ as a column ($n\times 1$ [[Matrix|matrix]]) then it is called a column vector. This is the convention. > - If write the elements of $\mathbb{R}^{n}$ as a row ($n\times 1$ matrix) then it is called a row vector. We may write $\mathbb{R}_{row}^{n}$ instead. > - We may write $(a_{1},a_{2},\dots,a_{n})^{T}$ to denote the column vector. > - The entries $a_{1},a_{2},\dots$ of a vector $\underline{v} = (a_{1},a_{2},\dot,a_{n})^{T}$ are called the components of $\underline{v}$. > - We say that $a_{i}$ is the $i$th component of $\underline{v}$. The components are real numbers. > - When giving names to vectors in $\mathbb{R}^{n}$, we usually underline them, writing $\underline{v} \in \mathbb{R}^{n}$ rather than $v \in \mathbb{R}^{n}$ > [!info] Remarks > - When drawing pictures, it is very useful to draw vectors as arrows. But remember that each vector is really that single point of $\mathbb{R}^{n}$ that sits at the tip of the arrow. > - Sometimes people refer to these as position vectors to distinguish them from, for example, velocity vectors or force vectors. # Properties - [[Dot Product in Real n-Space]]; [[Euclidean Norm]]; [[Angle Between Nonzero Real Vectors]]; [[Cauchy-Schwartz inequality]]; [[Absolute Value Satisfies Triangle Inequality]]; - [[Orthonormal Subset of Euclidean Space]]; [[Orthogonal Projection in Real n-Space]]; [[Gram-Schmidt orthogonalisation in real n-space]]. - [[Lines and planes in Real 3-Space]]. # Applications - [[Euclidean spaces]].