# Versor (physics) 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.

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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 (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:

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

## Versor of a non-zero vector

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