Dot Product

A Dot Product is a Real-Valued Vector Function that is based on the Summation of the Products between each of pair of Vector Elements.

• AKA: ·, Canonical Dot Product, Inner Dot Product, Scalar Product, Dot Product Operation, Dot Product Function, Inner Product.
• Context:
  • Input: two Equal Sized Vectors.
  • Output: a Non-Negative Real Number [0,infinity)
  • It can define a Dot Product Space.
  • It can be used as a Vector Distance Function.
  • It can be written as <x,y> or also as (x·y).
  • Its Output is related to the angle between the two Vectors.
    • If the vectors have a Normalized Distance (1.0) then the Metric is the Cosine of the Angle between the two Vectors.
  • It can be used to calculate the Vector Length Function by Squareroot(Dot Product(x,x)).
  • Its Output is Zero when the two Vectors are Orthogonal Vectors.
• Example(s):
  • <[1,1],[1,1]> = 1x1 + 1x1 = 2 (notice that the length is SQRT(2))
  • <[1,0],[1,1]> = 1x0 + 1x1 = 1
  • <[1,0],[0,1]> = 1x0 + 0x1 = 0 (the two vectors are Orthogonal)
  • ·([1,0],[0,1]) = 0.
  • ·([0,3],[4,0] = 5.
• Counter-Example(s):
  • a Scalar-Vector Multiplication Function.
• Counter-Example(s):
  • Vector Multiplication Operation.
  • Cross Product.
  • Matrix Product.
• See: Kernel Function, Hyperplane, Cosine Distance Metric, Vector Multiplication, Vector Length Function.

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References

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/dot_product Retrieved:2014-4-26.
  • In mathematics, the dot product, or scalar product (or sometimes inner product in the context of Euclidean space), is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation can be defined either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the magnitudes of the two vectors and the cosine of the angle between them. The name "dot product" is derived from the centered dot " · " that is often used to designate this operation; the alternative name "scalar product" emphasizes the scalar (rather than vectorial) nature of the result.

    In three-dimensional space, the dot product contrasts with the cross product of two vectors, which produces a pseudovector as the result. The dot product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions.

2009

• (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=scalar%20product
  • S: (n) scalar product, inner product, dot product (a real number (a scalar) that is the product of two vectors)

• http://planetmath.org/encyclopedia/DotProduct.html
  • Let [math]\displaystyle{ u=(u_1,u_2,\ldots,u_n) }[/math] and [math]\displaystyle{ v=(v_1,v_2,\ldots,v_n) }[/math] two vectors on [math]\displaystyle{ k^n }[/math] where [math]\displaystyle{ k }[/math] is a field (like $\mathbb{R}$ or $\mathbb{C}$).

    Then we define the dot product of the two vectors as: [math]\displaystyle{ u\cdot v=u_1v_1+u_2v_2+\cdots+u_nv_n. }[/math] Notice that [math]\displaystyle{ u\cdot v }[/math] is NOT a vector but a scalar (an element from the field $k$).

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