Dot Product

Domain	Definition
Aerospace	= scalar product. (references)
Electrical Engineering	A scalar quantity P defined by using cartesian co-ordinates a)in an n-dimensional euclidian space, by the sum of the products of each component ai of the first quantity and the corresponding component bi of the second; b)in a three-dimensional space, by the product of the magnitudes of the two quantities and the cosine of the angle phi between them. Source: European Union. (references)
Source: compiled by the editor from various references; see credits.

Specialty Definition: Dot product

In mathematics, the dot product is a binary operation which takes two vectorss and returns a scalar quantity. It is also known as the inner product or scalar product.

It is defined as:

where θ is the angle between the two vectors. Thus, the dot product of two perpendicular vectors is always zero. If a and b are both unit vectors (ie of length 1), the dot product simply gives the cosine of the angle between them. Thus, given two vectors, the angle between them can be found by rearranging the above formula:

The dot product is particularly used in resolution of forces. If b is a unit vector, then the dot product gives the projection of a in direction b. In mechanics, this gives the component of a force in that direction.

Work is the dot product of force and displacement.

Properties

The definition has the following consequences:

the dot product is commutative, i.e. a·b = b·a.
two non-zero vectors a and b are perpendicular if and only if a·b = 0
the dot product is bilinear, i.e. a·(rb + c) = r (a·b) + (a·c)

From these it follows directly that the dot product of two vectors a = [a₁ a₂ a₃] and b = [b₁ b₂ b₃] given in coordinates can be computed particularly easily:

a·b = a₁b₁ + a₂b₂ + a₃b₃

or, using matrix multiplication and treating the vectors as 1-by-3 matrices:

a·b = ab^T

where b^T denotes the transpose of the matrix b.

The dot product satisfies all the axioms of an inner product. In an abstract vector space, the notion of angle between the elements of the space can be defined in terms of the inner product.

Proof that the two forms of definition are equivalent

;That is, given :a·b = a₁b₁ + a₂b₂ + a₃b₃ ;derive :a·b = |a|.|b|.cos(θ), ;or using nomenclature x for |x| :a·b = a.b.cos(θ).

This proof is shown for 3-dimensional vectors, but is readily extendable to N-dimensional vectors given mutually perpendicular unit vectors.

;Consider a vector :v =v₁i + v₂j + v₃k. ;Repeated application of the Pythagorean theorem determines that :v² =(v₁² + v₂² + v₃²). ;which is the same formula for the dot product of the vector v with itself, thus :v·v = v².

;Now consider two vectors a and b from the origin and separated by an angle θ. A third vector c may be defined as :c = a-b. ;Using the law of cosines, we have :c² = a² + b² - 2.a.b.cos(θ). ;And substituting the dot product for the squared lengths, we get :c·c = a·a + b·b - 2.a.b.cos(θ). ;But as c = a - b, we also have : c·c = (a - b)·(a - b). ;which expands to :c·c = a·a + b·b - 2.a·b. ;Then merging the two c·c equations we obtain :a·a + b·b - 2.a·b = a·a + b·b - 2.a.b.cos(θ). ;Subtracting a·a + b·b from both sides leaves :- 2.a·b = - 2.a.b.cos(θ). ;And dividing by -2 derives the final :a·b = a.b.cos(θ).

See also: Cross product

Frequency of Internet Keywords: Dot Product

Modern Translation: Dot Product

Language		Translations for "dot product"; alternative meanings/domain in parentheses.
Danish		skalarprodukt (scalar product). (various references)

Dutch		scalarprodukt (scalar product), scalair produkt (scalar product). (various references)

Finnish		skalaaritulo (scalar product), pistetulo (scalar product). (various references)

French		produit scalaire. (various references)

German		Skalarprodukt (scalar product), skalares Produkt (scalar product). (various references)

Greek		εσωτερικό γινόμενο (scalar product). (various references)

Italian		prodotto scalare (scalar product). (various references)

Pig Latin		otday oductpray

Portuguese		produto escalar (scalar product). (various references)

Spanish		producto escalar (scalar product). (various references)

Swedish		skalär produkt (scalar product). (various references)
Source: compiled by the editor from various translation references.

Anagrams: Dot Product

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "c-d-d-o-o-p-r-t-t-u"
	-3 letters: dropout, outcrop, outdrop, outport, product.
	-4 letters: copout, doctor, prutot, uproot, uropod.
	-5 letters: coopt, court, croup, droop, dropt, duroc, odour, outdo, potto, proud, putto, torot, troop, trout, tutor.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Alternative Orthography: Dot Product

Hexadecimal (or equivalents, 770AD-1900s) (references)

44 6F 74 50 72 6F 64 75 63 74

Leonardo da Vinci (1452-1519; backwards) (references)

Binary Code (1918-1938, probably earlier) (references)

01000100 01101111 01110100 00100000 01010000 01110010 01101111 01100100 01110101 01100011 01110100

HTML Code (1990) (references)

&#68 &#111 &#116 &#32 &#80 &#114 &#111 &#100 &#117 &#99 &#116

ISO 10646 (1991-1993) (references)

0044 006F 0074 0050 0072 006F 0064 0075 0063 0074

Encryption (beginner's substitution cypher): (references)

Expression	Frequency per Day
dot product	31
dot product vector	8
Source: compiled by the editor from various references; see credits.

Dot Product

Definition: Dot Product

Dot Product

Noun

Specialty Definition: Dot Product

Aerospace

Electrical Engineering

Specialty Definition: Dot product

Properties

Proof that the two forms of definition are equivalent

Synonyms: Dot Product

Frequency of Internet Keywords: Dot Product

dot product

dot product vector

Modern Translation: Dot Product

Danish

Dutch

Finnish

French

German

Greek

Italian

Pig Latin

Portuguese

Spanish

Swedish

Anagrams: Dot Product

Scrabble® Enable2K-Verified Anagrams

Alternative Orthography: Dot Product

44 6F 74 50 72 6F 64 75 63 74

01000100 01101111 01110100 00100000 01010000 01110010 01101111 01100100 01110101 01100011 01110100

&#68 &#111 &#116 &#32 &#80 &#114 &#111 &#100 &#117 &#99 &#116

0044 006F 0074 0050 0072 006F 0064 0075 0063 0074

388186250848170876986

INDEX