Versor (physics)

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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 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., 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}.