7. Norm

A scalar that measures the length (or magnitude) of a vector.

Notation: .

7.1. Euclidean norm ()

The default norm — Pythagoras applied componentwise.

For :

7.2. From dot product

The Euclidean norm is the dot product of a vector with itself, square-rooted:

(See Dot Product.)

This connection makes norms central to projections, angles between vectors, and least-squares regression.

7.3. Properties (norm axioms)

For any vectors and scalar :

• Non-negativity:
• Definiteness:
• Absolute homogeneity:
• Triangle inequality: (see Triangle Inequality)

Any function satisfying these four axioms is a norm.

7.4. Geometric interpretation

• = distance from the origin to the point
• = distance between the two points and
• The set is the unit sphere (a circle in , sphere in )

7.5. Other common norms ( family)

For :

Special cases:

7.6. Normalizing a vector

Dividing a vector by its norm gives a unit vector in the same direction:

7.7. Connections

• Dot Product —
• Cauchy–Schwarz Inequality —
• Triangle Inequality —
• Unit Vector — vectors with
• Angles Between Vectors —