# Versor (physics)

In geometry and physics, the **versor** of an axis or of a vector is a unit vector indicating its direction.

The versor of a Cartesian axis is also known as a **standard basis vector**. The versor of a vector is also known as a **normalized vector**.

## Versors of a Cartesian coordinate system

The versors of the axes of a Cartesian coordinate system are the unit vectors codirectional with the axes of that system. Every Euclidean vector **a** in a *n*-dimensional Euclidean space (**R**^{n}) can be represented as a linear combination of the *n* versors of the corresponding Cartesian coordinate system. For instance, in a three-dimensional space (**R**^{3}), there are three versors:

They indicate the direction of the Cartesian axes *x*, *y*, and *z*, respectively. In terms of these, any vector **a** can be represented as

where **a**_{x}, **a**_{y}, **a**_{z} are called the vector components (or vector projections) of **a** on the Cartesian axes *x*, *y*, and *z* (see figure), while *a*_{x}, *a*_{y}, *a*_{z} are the respective scalar components (or scalar projections).

In linear algebra, the set formed by these *n* versors is typically referred to as the standard basis of the corresponding Euclidean space, and each of them is commonly called a **standard basis vector**.

### Notation

A hat above the symbol of a versor is sometimes used to emphasize its status as a unit vector (e.g., ).

In most contexts it can be assumed that **i**, **j**, and **k**, (or and ) are versors of a 3-D Cartesian coordinate system. The notations , , , or , with or without hat, are also used, particularly in contexts where **i**, **j**, **k** might lead to confusion with another quantity. This is recommended, for instance, when index symbols such as *i*, *j*, *k* are used to identify an element of a set of variables.

## Versor of a non-zero vector

The versor (or **normalized vector**) of a non-zero vector is the unit vector codirectional with :

where is the norm (or length) of .