Unit vector – What it is, definition and concept

A unit vector or normalized vector is a vector that has direction and sense, has no dimension and its magnitude or module is equal to one.

In other words, a unit vector is a vector that has direction and sense with a magnitude equal to one but is dimensionless.

Modulus of a vector

The modulus of a vector is the magnitude of a segment oriented in a space that is determined by two points and their order. Simplifying, the module of a vector is the length between the beginning and the end of the vector, that is, where the arrow begins and where it ends.

Given a two-dimensional vector v with coordinates (v1, v2), the module would be such that:

Module Of A Vector
Modulus of a vector


The main characteristic of a unit vector is that the module is equal to one and that it is dimensionless. The property of dimensionlessness is due to the fact that this vector does not have a preset dimension by itself.

This characteristic is present in the normalization of a vector where it is intended to find a vector with the same sense and direction with a module equal to one (unit vector) from a given non-zero vector.

Therefore, as this unit vector will depend on the dimensions of the reference vector, the unit vector will adapt and take on these dimensions. Consequently, we will speak of adimensionality because previously there is no dimension established until it is determined by the reference vector.


The unit vector is also called a normalized vector by the action of normalize a vector. This process consists of finding a vector with the same direction and sense as the vector that we want to normalize, but especially with a module equal to 1.

In other words, we start from a non-zero vector and replicate its direction and sense, but keeping its modulus equal to one. The final result is two vectors: the original vector and the unit vector with the same direction and sense as the original vector, but with a module equal to one.


It is common to find the unit vector expressed by a letter and with a caret, of the form: â. It can also be represented by the Greek letter mu and a subscript that indicates the vector whose direction and sense is replicated.

Unit Vector
Unit vector
Unit Vector Relative To A Vector
Unit vector with respect to a vector
Cartersian Three-Dimensional Plane
Three-dimensional Cartesian plane with the unit vector and its reference vector

In the previous three-dimensional Cartesian plane you can see the unit vector and any non-zero vector. We can see that the unit vector has the same direction and sense as the vector to, but a different module.


Determine if the following vector modules belong to unit vectors:

Vector Modules
Modules of a vector

The only module that belongs to a unit vector is the last module that corresponds to vector d.

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