Vector Length Function

A Vector Length Function is a length function that is restricted to vectors.

• AKA: Vector Magnitude, Vector Length, Norm Function, Vector Norm, Vector Norm Function.
• Context:
  • It can be written as ||x||
• Example(s):
  • ||[1,1]|| = SQRT(2)
• Counter-Example(s)
• a Vector Direction.
• See: Dot Product, Scalar, Vector Normalization Function, Normed Vector Space.

---

---

References

2009

• http://en.wikipedia.org/wiki/Norm_(mathematics)
  • In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector. A seminorm (or pseudonorm), on the other hand, is allowed to assign zero length to some non-zero vectors.
  • A simple example is the 2-dimensional Euclidean space R2 equipped with the Euclidean norm. Elements in this vector space (e.g., (3, 7) ) are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at the origin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude.
  • A vector space with a norm is called a normed vector space. Similarly, a vector space with a seminorm is called a seminormed vector space.
• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Dot_product#Geometric_interpretation
  • In Euclidean geometry, the dot product, length, and angle are related. For a vector a, the dot product a · a is the square of the length of a, or
    • |\mathbf{a}| = \sqrt{\mathbf{a} ∙ \mathbf{a}}
  • where |a| denotes the length (magnitude) of a. More generally, if b is another vector
    • \mathbf{a} ∙ \mathbf{b} = |\mathbf{a}| \, |\mathbf{b}| \cos θ \,

---