By Kardi Teknomo, PhD.

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Vector Norm

Based on Pythagorean Theorem, the vector from the origin to the point (3, 4) in 2D Euclidean plane has length of Vector Norm and the vector from the origin to the point Vector Norm has lengthVector Norm. The length of a vector with two elements is the square root of the sum of each element squared.

The magnitude of a vector is sometimes called the length of a vector, or norm of a vector. Basically, norm of a vector is a measure of distance, symbolized by double vertical barVector Norm.
The magnitude of a vector can be extended to Vector Normdimensions. A vector a with Vector Norm elements has length
Vector Norm

The vector length is called Euclidean length or Euclidean norm. Mathematician often used term norm instead of length. Vector norm is defined as any function that associated a scalar with a vector and obeys the three rules below

  1. Norm of a vector is always positive or zeroVector Norm. The norm of a vector is zero if and only if the vector is a zero vectorVector Norm.
  2. A scalar multiple to a norm is equal to the product of the absolute value of the scalar and the normVector Norm.
  3. Norm of a vector obeys triangular inequality that the norm of a sum of two vectors is less than or equal to the sum of the normsVector Norm.

There are many common norms:

  • 1-norm is defined by the sum of absolute value of the vector elements
    Vector Norm.
  • 2-norm is the most often used vector norm, sometimes called Euclidean norm. When the subscript index of the vector norm is not specified, you may think that it is a Euclidean normVector Norm.
  • p-norm is sometimes called Minskowski norm is defined asVector Norm . p-norm is generalized norm with a parameter Vector Norm.
  • max-norm is also called Chebyshev norm is the largest absolute element in the vector Vector Norm.

Use the interactive program below to experiment with your own vector input. The program will give you the norm of vector for p=1, 2, 3 and max. The vector input will be redrawn to give you feedback on what you type. Click Random Example button to generate random vector.



Some important properties of vector norm are

  • Square of Euclidean norm is equal to the sum of squareVector Norm.
  • Pythagorean Theorem is hold if and only if the two vectors are orthogonalVector Norm.
  • The law of cosineVector Norm
  • Norm of the dot product of two vectors is equal to the product of their normsVector Norm.
  • Relationship to vector inner products
    • Square of Euclidean norm of a vector is equal to the inner product to itself Vector Norm
    • Vector Norm where, Vector Norm is the angle between the two vectors
    • Vector Norm
  • Norm of addition or subtraction follow the law of cosine Vector Norm
  • Addition of two square of norm of vectors follow parallelogram law Vector Norm
  • p-norm is greater than the max-norm but less than Vector Norm times the max-norm, that is Vector Norm.
  • The norm ratio satisfies the inequalityVector Norm. As Vector Normtends to infinity, the Vector Normapproaches Vector NormandVector Norm approachesVector Norm.
  • Cauchy-Schwartz inequality stated that the absolute value of vector dot product is always less than or equal to the product of their normsVector Norm. The equality Vector Normholds if and only if the vectors are linearly dependent.
  • Relationship of norm of cross product and dot product isVector Norm.

See Also: Similarity tutorial, Minskowski distance, Chebyshev distance, Euclidean distance, City Block distance, Resources on Linear Algebra

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Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. https:\\\kardi\tutorial\LinearAlgebra\