A **vector** is simply an element of a [[vector space]] A Euclidean **vector** is an object that has magnitude and direction A vector can also be thought of a [[tuple]] of [[components]], and a corresponding vector-space. The number of these necessary components is the dimension of the vector-space, and sometimes it is also called the [[dimension of a vector|dimension]] of the vector. If $\boldsymbol v \in V$ is an $n$-tuple of $v_i \in F$, then $\boldsymbol v \in F^n$ $\boldsymbol{v}=\left[\begin{array}{l}v_{1} \\ v_{2}\end{array}\right] \quad \begin{array}{l}v_{1}=\text { first component of } \boldsymbol{v} \\ v_{2}=\text { second component of } \boldsymbol{v}\end{array}$ --- When someone typing math is on their best behavior, they use bold variables for vectors, like $\boldsymbol{v}$. When handwriting, the negative effect of bold variables on pencils and wrists leads most people to notate vectors with an arrow over them instead, like $\overrightarrow{v}$. Sometimes these arrows make it into print, even through they are harder to type. Still, it is even possible to find examples of bold variables with arrows over them, like $\boldsymbol{\overrightarrow{v}}$, in its 29-characters of $\LaTeX$ beauty