# Definition:Zero Vector

## Definition

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

Let $\struct {G, +_G, \circ}_R$ be an $R$-module.

The identity of $\struct {G, +_G}$ is usually denoted $\bszero$, or some variant of this, and called the zero vector:

$\forall \mathbf a \in \struct {G, +_G, \circ}_R: \bszero +_G \mathbf a = \mathbf a = \mathbf a +_G \bszero$

Note that on occasion it is advantageous to denote the zero vector differently, for example by $e$, or $\bszero_V$ or $\bszero_G$, in order to highlight the fact that the zero vector is not the same object as the zero scalar.

### Zero Vector in $\R^n$

Let $\struct {\R^n, +, \times}_\R$ be a real vector space.

The zero vector in $\struct {\R^n, +, \times}_\R$ is:

$\mathbf 0_{n \times 1} := \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$

where $0 \in \R$.

### Zero Vector Quantity

A vector quantity whose magnitude is zero is referred to as a zero vector.

## Also known as

A zero vector is also seen referred to as the null vector.

Some sources refer to the neutral element.

The term origin is sometimes seen, but this has a more precise definition in the context of analytic geometry, and so its use is not recommended here.

## Also see

• Results about zero vectors can be found here.