#
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:

- $\mathbf {i} =(1,0,0),$
- $\mathbf {j} =(0,1,0),$
- $\mathbf {k} =(0,0,1).$

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

- $\mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}\mathbf {i} +a_{y}\mathbf {j} +a_{z}\mathbf {k} ,$

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., ${\hat {\mathbf {\imath } }}$).

In most contexts it can be assumed that **i**, **j**, and **k**, (or ${\vec {\imath }},$ ${\vec {\jmath }},$ and ${\vec {k}}$) are versors of a 3-D Cartesian coordinate system. The notations $({\hat {\mathbf {x} }},{\hat {\mathbf {y} }},{\hat {\mathbf {z} }})$, $({\hat {\mathbf {x} }}_{1},{\hat {\mathbf {x} }}_{2},{\hat {\mathbf {x} }}_{3})$, $({\hat {\mathbf {e} }}_{x},{\hat {\mathbf {e} }}_{y},{\hat {\mathbf {e} }}_{z})$, or $({\hat {\mathbf {e} }}_{1},{\hat {\mathbf {e} }}_{2},{\hat {\mathbf {e} }}_{3})$, 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**) ${\hat {\mathbf {u} }}$ of a non-zero vector $\mathbf {u}$ is the unit vector codirectional with $\mathbf {u}$:

- ${\hat {\mathbf {u} }}={\frac {\mathbf {u} }{\|\mathbf {u} \|}}.$

where $\|\mathbf {u} \|$ is the norm (or length) of $\mathbf {u}$. Notice that a versor lost the units of the original vector. For instance, if we have the vector ${\vec {u}}=(0,5,0)\,\mathrm {m}$, then $|{\vec {u}}|={\sqrt {0^{2}+5^{2}+0^{2}}}\,\mathrm {m} =5\,\mathrm {m}$ and

${\hat {u}}={\frac {\vec {u}}{|{\vec {u}}|}}={\frac {(0,5,0)\,\mathrm {m} }{5\,\mathrm {m} }}=(0,1,0)$

You can notice that ${\hat {u}}$ is a dimensionless quantity.

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