Versor (physics)

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:

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

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., ı ^ ).

In most contexts it can be assumed that i, j, and k, (or ı → , ȷ → , and k → ) are versors of a 3-D Cartesian coordinate system. The notations ( x ^ , y ^ , z ^ ) , ( x ^ 1 , x ^ 2 , x ^ 3 ) , ( e ^ x , e ^ y , e ^ z ) , or ( e ^ 1 , e ^ 2 , 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) u ^ of a non-zero vector u is the unit vector codirectional with u :

where ∥ u ∥ is the norm (or length) of u .