# 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 i, j, k of the Cartesian axes x, y, z for a three-dimensional Euclidean space. Every vector a in that space is a linear combination of these versors.

## 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 (Rn) can be represented as a linear combination of the n versors of the corresponding Cartesian coordinate system. For instance, in a three-dimensional space (R3), there are three versors:

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

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

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

where ax, ay, az are called the vector components (or vector projections) of a on the Cartesian axes x, y, and z (see figure), while ax, ay, az 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., ${\displaystyle {\hat {\mathbf {\imath } }}}$).

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

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

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

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

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

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