Physics:Versor

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.

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 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., $\hat{\mathit{ı}}$).

In most contexts it can be assumed that i, j, and k, (or $\overrightarrow{ı},$ $\overrightarrow{ȷ},$ and $\overrightarrow{k}$) are versors of a 3-D Cartesian coordinate system. The notations $(\widehat{\mathbf{𝐱}},\widehat{\mathbf{𝐲}},\widehat{\mathbf{𝐳}})$, $({\widehat{\mathbf{𝐱}}}_1,{\widehat{\mathbf{𝐱}}}_2,{\widehat{\mathbf{𝐱}}}_3)$, $({\widehat{\mathbf{𝐞}}}_x,{\widehat{\mathbf{𝐞}}}_y,{\widehat{\mathbf{𝐞}}}_z)$, or $({\widehat{\mathbf{𝐞}}}_1,{\widehat{\mathbf{𝐞}}}_2,{\widehat{\mathbf{𝐞}}}_3)$, 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) $\widehat{\mathbf{𝐮}}$ of a non-zero vector $\mathbf{𝐮}$ is the unit vector codirectional with $\mathbf{𝐮}$:

$\widehat{\mathbf{𝐮}}=\frac{\mathbf{𝐮}}{\Vert \mathbf{𝐮}\Vert }.$

where $\Vert \mathbf{𝐮}\Vert$ 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 $\overrightarrow{u}=(0,5,0)\mathrm{m}$, then $|\overrightarrow{u}|=\sqrt{0^2+5^2+0^2}\mathrm{m}=5\mathrm{m}$ and

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

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