represents the domain of vectors of dimension d.


represents the domain of vectors of dimension d, with components in the domain dom.


  • A valid dimension specification d in Vectors[d,dom] is any positive integer. It is also possible to work with symbolic dimension specifications.
  • A valid component domain specification dom in Vectors[d,dom] is either Reals or Complexes.
  • The domain Vectors[d] is automatically converted into Vectors[d,Complexes].


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Basic Examples  (1)

A vector in dimension 3:

Properties of tensor products of the vector:

An identity valid for any three-dimensional vector:

Scope  (3)

Declare vectors of any dimension:

Compute properties of tensors formed from those vectors:

Work with vectors in symbolic dimension:

A real vector:

Applications  (4)

Declare several objects as vectors:

Extract their properties:

Check identities:

Check whether a vector belongs to a given domain:

Conditions involving symbolic parameters may be converted into simpler conditions:

Check a subdomain relation:

Check vector identities:

Commutativity of the dot product with vectors:

Properties & Relations  (3)

Vectors can also be defined using Arrays with rank 1. These two assumptions are equivalent:

Vectors cannot contain other lists:

Two alternative ways of checking numerical vectors:

Possible Issues  (4)

Addition of symbolic and explicit vectors is determined by the Listable attribute of Plus:

Hence, listability will in general affect operations that simultaneously involve both symbolic and explicit vectors.

The zero vector may be represented as 0 in symbolic computations:

Dimension 0 is not accepted:

{} is interpreted in a special way in Element, such that it returns True irrespectively of the domain used:

Wolfram Research (2012), Vectors, Wolfram Language function,


Wolfram Research (2012), Vectors, Wolfram Language function,


Wolfram Language. 2012. "Vectors." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2012). Vectors. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_vectors, author="Wolfram Research", title="{Vectors}", year="2012", howpublished="\url{}", note=[Accessed: 27-May-2024 ]}


@online{reference.wolfram_2024_vectors, organization={Wolfram Research}, title={Vectors}, year={2012}, url={}, note=[Accessed: 27-May-2024 ]}