Vector Length Function

A Vector Length Function is a vector-input scalar-output function that is a length function (i.e.is restricted to vectors from some vector space).

• AKA: Vector Magnitude, Vector Norm.
• Context:
  • It can (often) be written as [math]\displaystyle{ \left\|\mathbf{a}\right\| }[/math].
  • It can (typically) be a Convex Function.
• Example(s):
  • ‖[1,1]‖ = SQRT(2)
  • a Word Vector Length Function.
  • …
• Counter-Example(s)
  • a Vector Direction.
  • a Surface Area.
• See: Dot Product, Scalar, Vector Normalization Function, Normed Vector Space, Pythagorean Theorem.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Euclidean_vector#Length Retrieved:2015-1-9.
  • The length or magnitude or norm of the vector a is denoted by ‖a‖ or, less commonly, |a|, which is not to be confused with the absolute value (a scalar "norm").

    The length of the vector a can be computed with the Euclidean norm  :[math]\displaystyle{ \left\|\mathbf{a}\right\|=\sqrt{{a_1}^2+{a_2}^2+{a_3}^2} }[/math]

    which is a consequence of the Pythagorean theorem since the basis vectors e₁, e₂, e₃ are orthogonal unit vectors.

    This happens to be equal to the square root of the dot product, discussed below, of the vector with itself:  :[math]\displaystyle{ \left\|\mathbf{a}\right\|=\sqrt{\mathbf{a}\cdot\mathbf{a}}. }[/math]

    ;Unit vector

    A unit vector is any vector with a length of one; normally unit vectors are used simply to indicate direction. A vector of arbitrary length can be divided by its length to create a unit vector. This is known as normalizing a vector. A unit vector is often indicated with a hat as in â.

    To normalize a vector a = [a₁, a₂, a₃], scale the vector by the reciprocal of its length ||a||. That is:  :[math]\displaystyle{ \mathbf{\hat{a}} = \frac{\mathbf{a}}{\left\|\mathbf{a}\right\|} = \frac{a_1}{\left\|\mathbf{a}\right\|}\mathbf{e}_1 + \frac{a_2}{\left\|\mathbf{a}\right\|}\mathbf{e}_2 + \frac{a_3}{\left\|\mathbf{a}\right\|}\mathbf{e}_3 }[/math]

    ;Null vector

    The null vector (or zero vector) is the vector with length zero. Written out in coordinates, the vector is (0,0,0), and it is commonly denoted [math]\displaystyle{ \vec{0} }[/math], or 0, or simply 0. Unlike any other vector it has an arbitrary or indeterminate direction, and cannot be normalized (that is, there is no unit vector which is a multiple of the null vector). The sum of the null vector with any vector a is a (that is, 0+a=a).

2009a

• (Wikipedia, 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.

2009b

• (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 $a \cdot a$ is the square of the length of a, or
    • $|\mathbf{a}| = \sqrt{\mathbf{a} \cdot \mathbf{a}}$
  • where $|a|$ denotes the length (magnitude) of $a$. More generally, if b is another vector
    • $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \, |\mathbf{b}| \cos \theta \, $

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