unit vector (original) (raw)

A unit vector is a unit-length element of Euclidean space. Equivalently, one may say that the norm of a unit vector is equal to 1, and write ∥𝐮∥=1, where 𝐮 is the vector in question.

Let 𝐯 be a non-zero vector. To normalize 𝐯 is to find the unique unit vector with the same direction as 𝐯. This is done by multiplying 𝐯 by the reciprocal of its length; the corresponding unit vector is given by 𝐮=𝐯∥𝐯∥.

Note:

The concept of a unit vector and normalization makes sense in any vector space equipped with a real or complex norm. Thus, in quantum mechanics one represents states as unit vectors belonging to a (possibly) infinite-dimensional Hilbert space. To obtain an expression for such states one normalizes the results of a calculation.

Example:

Consider ℝ3 and the vector𝐯=(1,2,3). The norm (length) is 14. Normalizing, we obtain the unit vector 𝐮 pointing in the same direction, namely𝐮=(114,214,314).