---

Vector

a.k.a.

• Column vector
• Point

Define

… in terms of scalars as the member of a vector space

\[\begin{equation*}
  \vec{v}
  = \begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix}
  \in V^n,
\end{equation*}
\]

given

\[\begin{equation*}
  \forall i : v_i \in F, \quad
\end{equation*}
\]

where	\(v_i\)	is the \(i\)-th scalar component of \(\vec{v}\),
	\(V^n\)	is the vector space with \(\vec{v}\), and
	\(F\)	is the field over which \(V^n\) is defined.

Imagine

… on the real plane as either

• the point \(P = (v_1, v_2)\) or
• the line segment from the origin to \(P\).

Denote

… as shown above, or with the square brackets

\[\begin{equation*}
  \vec{v} = \begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix}.
\end{equation*}
\]