Unit vector - Knowlpedia

A unit vector in a normed vector space is a vector $v$ satisfying $\|v\|=1$ . If $w\ne0$ , then

\widehat w=\frac{w}{\|w\|}

is a unit vector. Normalization is undefined at the zero vector.

Euclidean interpretation

In a real inner-product space, the condition is $v\cdot v=1$ . Unit vectors identify directions; an orthonormal basis consists of unit vectors that are pairwise perpendicular. The set of all unit vectors is the unit sphere.

References

Sheldon Axler, Linear Algebra Done Right, 4th ed., Chapters 2–3, 6 and 9.