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.

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., $unit vector i$).

In most contexts it can be assumed that i, j, and k, (or $vector i$ $vector j$ and $vector k$) are versors of a 3-D Cartesian coordinate system. The notations $x-hat, y-hat, z-hat$, $x-hat sub 1, x-hat sub 2, x-hat sub 3$, $e-hat sub x, e-hat sub y, e-hat sub z$, or $e-hat sub 1, e-hat sub 2, e-hat sub 3$, 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 $\mathbf{u}$.