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, although that term has a more specialised meaning in Riemannian geometry
- the trivial vector
- the neutral element, but that term is also used for the more general concept of an identity element.
The term origin is sometimes seen, but this has a more precise definition in the context of analytic geometry.
Care may therefore need to be taken not to confuse these concepts.
Also see
- Results about zero vectors can be found here.
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 4$. Vector Spaces
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.2$. Vector Spaces
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space
- 1971: Kenneth Hoffman and Ray Kunze: Linear Algebra (2nd ed.): Chapter $2$: Vector Spaces : $\S 2.1$: Vector Spaces
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): zero vector
- 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.2$ Vector Spaces over $C$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): zero vector (null vector): 2.