Definition:Zero Vector

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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

The zero vector is also sometimes known 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.