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Encyclopedia > Dot product

In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... Please refer to Real vs. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

Contents

Definition and examples

The dot product of two vectors (from an orthonormal vector space) a = [a1, a2, … , an] and b = [b1, b2, … , bn] is by definition: In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

mathbf{a}cdot mathbf{b} = sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + cdots + a_nb_n

where Σ denotes summation notation. Summation is the addition of a set of numbers; the result is their sum. ...


For example, the dot product of two three-dimensional vectors [1, 3, −5] and [4, −2, −1] is

begin{bmatrix}1&3&-5end{bmatrix} cdot begin{bmatrix}4&-2&-1end{bmatrix} = (1)(4) + (3)(-2) + (-5)(-1) = 3.

Using matrix multiplication and treating the (column) vectors as n×1 matrices, the dot product can also be written as: This article gives an overview of the various ways to perform matrix multiplication. ... For the square matrix section, see square matrix. ...

mathbf{a} cdot mathbf{b} = mathbf{a}^T mathbf{b} ,

where aT denotes the transpose of the matrix a. Using the example from above, this would result in a 1×3 matrix (i.e., vector) multiplied by a 3×1 vector (which, by virtue of the matrix multiplication, results in a 1×1 matrix, i.e., a scalar): In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

begin{bmatrix} 1&3&-5 end{bmatrix}begin{bmatrix} 4-2-1 end{bmatrix} = begin{bmatrix} 3 end{bmatrix}.

Geometric interpretation

In the Euclidean space there is a strong relationship between the dot product and lengths and angles. For a vector a, aa is the square of its length, and, more generally, if b is another vector Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... Length is the long dimension of any object. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...

mathbf{a} cdot mathbf{b} = |mathbf{a}| , |mathbf{b}| cos theta ,

where |a| and |b| denote the length (magnitude) of a and b, and θ is the angle between them. Length is the long dimension of any object. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...


Since |a|•cos(θ) is the scalar projection of a onto b, the dot product can be understood geometrically as the product of this projection with the length of b. The scalar resolute of two vectors, in the direction of (also on ), is given by: or (where is the angle between vectors and ). The scalar resolute is a scalar, and represents the length of the vector mapped onto vector . ...


As the cosine of 90° is zero, the dot product of two perpendicular vectors is always zero. If a and b have length one (they are unit vectors), 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: In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Fig. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

theta = arccos left( frac {bold{a}cdotbold{b}} {|bold{a}||bold{b}|}right).

Sometimes these properties are also used for defining the dot product, especially in 2 and 3 dimensions; this definition is equivalent to the above one. For higher dimensions the formula can be used to define the concept of angle.


The geometric properties rely on the basis of vectors being perpendicular and having unit length: either we start with such a basis, or we use an arbitrary basis and define length and angle (including perpendicularity) with the above. In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...


As the geometric interpretation shows, the dot product is invariant under isometric changes of the basis: rotations, reflections, and combinations, keeping the origin fixed.


In other words, and more generally for any n, the dot product is invariant under a coordinate transformation based on an orthogonal matrix. This corresponds to the following two conditions: See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...

  • the new basis is again orthonormal (i.e. it is orthonormal expressed in the old one)
  • the new base vectors have the same length as the old ones (i.e. unit length in terms of the old basis)

The dot product in physics

In physics, magnitude is a scalar in the physical sense, i.e. a physical quantity independent of the coordinate system, expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. The formula in terms of coordinates is evaluated with not just numbers, but numbers times units. Therefore, although it relies on the basis being orthonormal, it does not depend on scaling. Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ... In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... A physical quantity is either a quantity within physics that can be measured (e. ... A number is an abstract entity that represents a count or measurement. ... The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. ...


Example:

Mechanical work is a force applied through a distance, defined mathematically as the line integral of a scalar product of force and displacement vectors. ... In physics, force is an influence that may cause a body to accelerate. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ...

Properties

The following properties hold if a, b, and c are vectors and r is a scalar. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...


The dot product is commutative: In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

mathbf{a} cdot mathbf{b} = mathbf{b} cdot mathbf{a}.

The dot product is bilinear: In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

mathbf{a} cdot (rmathbf{b} + mathbf{c}) = r(mathbf{a} cdot mathbf{b}) +(mathbf{a} cdot mathbf{c}).

The dot product is distributive: In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

mathbf{a} cdot (mathbf{b} + mathbf{c}) = mathbf{a} cdot mathbf{b} + mathbf{a} cdot mathbf{c}.

When multiplied by a scalar value, dot product satisfies:

(c_1mathbf{a}) cdot (c_2mathbf{b}) = (c_1c_2) (mathbf{a} cdot mathbf{b})

(these last two properties follow from the first two).


Two non-zero vectors a and b are perpendicular if and only if ab = 0. Fig. ... It has been suggested that this article or section be merged with Logical biconditional. ...


If b is a unit vector, then the dot product gives the magnitude of the projection of a in the direction b, with a minus sign if the direction is opposite. Decomposing vectors is often useful for conveniently adding them, e.g. in the calculation of net force in mechanics. In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... This article is about vectors. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ...


Unlike multiplication of ordinary numbers, where if ab = ac, then b always equals c unless a is zero, the dot product does not obey the cancellation law: In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a * b = a * c always implies b = c. ...

  • If ab = ac and a0:
  • Then we can write: a • (b - c) = 0 by the distributive law; and from the previous result above:
  • If a is perpendicular to (b - c), we can have (b - c) ≠ 0 and therefore bc.

In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

Generalization

The inner product generalizes the dot product to abstract vector spaces, it is normally denoted by 〈a, b〉. Due to the geometric interpretation of the dot product the norm ||a|| of a vector a in such an inner product space is defined as In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

|mathbf{a}| = sqrt{langlemathbf{a}, mathbf{a}rangle},

such that it generalizes length, and the angle θ between two vectors a and b by

cos{theta} = frac{langlemathbf{a}, mathbf{b}rangle}{|mathbf{a}| , |mathbf{b}|}.

In particular, two vectors are considered orthogonal if their dot product is zero In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

The Frobenius inner product defines an inner product on matrices as though they are two-dimensional vectors, summing up the products of corresponding components. This article gives an overview of the various ways to perform matrix multiplication. ...


Proof of the geometric interpretation

Note: This proof is shown for 3-dimensional vectors, but is readily extendable to n-dimensional vectors.


Consider a vector

mathbf{v} = v_1 mathbf{i} + v_2 mathbf{j} + v_3 mathbf{k}. ,

Repeated application of the Pythagorean theorem yields for its length v In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...

v^2 = v_1^2 + v_2^2 + v_3^2. ,

But this is the same as

mathbf{v} cdot mathbf{v} = v_1^2 + v_2^2 + v_3^2, ,

so we conclude that taking the dot product of a vector v with itself yields the squared length of the vector.

Lemma 1
mathbf{v} cdot mathbf{v} = v^2. ,

Now consider two vectors a and b extending from the origin, separated by an angle θ. A third vector c may be defined as In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ...

mathbf{c}  stackrel{mathrm{def}}{=} mathbf{a} - mathbf{b}. ,

creating a triangle with sides a, b, and c. According to the law of cosines, we have Fig. ...

c^2 = a^2 + b^2 - 2 ab cos theta. ,

Substituting dot products for the squared lengths according to Lemma 1, we get

mathbf{c} cdot mathbf{c} = mathbf{a} cdot mathbf{a} + mathbf{b} cdot mathbf{b} - 2 ab costheta. ,                   (1)

But as cab, we also have

mathbf{c} cdot mathbf{c} = (mathbf{a} - mathbf{b}) cdot (mathbf{a} - mathbf{b}) ,,

which, according to the distributive law, expands to In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

mathbf{c} cdot mathbf{c} = mathbf{a} cdot mathbf{a} + mathbf{b} cdot mathbf{b} -2(mathbf{a} cdot mathbf{b}). ,                     (2)

Merging the two cc equations, (1) and (2), we obtain

mathbf{a} cdot mathbf{a} + mathbf{b} cdot mathbf{b} -2(mathbf{a} cdot mathbf{b}) = mathbf{a} cdot mathbf{a} + mathbf{b} cdot mathbf{b} - 2 ab costheta. ,

Subtracting aa + bb from both sides and dividing by −2 leaves

mathbf{a} cdot mathbf{b} = ab costheta. ,

Q.E.D. Q.E.D. is an abbreviation of the Latin phrase (literally, which was to be demonstrated). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven. ...


See also

In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ... In mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Bunyakovski–Schwarz inequality, named after Augustin Louis Cauchy, Viktor Yakovlevich Bunyakovsky and Hermann Amandus Schwarz, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in...

External links


  Results from FactBites:
 
Dot product - Wikipedia, the free encyclopedia (996 words)
In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity.
As the cosine of 90° is zero, the dot product of two perpendicular vectors is always zero.
The inner product generalizes the dot product to abstract vector spaces, it is normally denoted by .
Product (mathematics) - Wikipedia, the free encyclopedia (186 words)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.
When matrices or members of various other associative algebras are multiplied the product usually depends on the order of the factors; in other words, matrix multiplication, and the multiplications in those other algebras, are non-commutative.
The dot product and cross product are forms of multiplication of vectors; the same as dot product or more general are the scalar product and the inner product; see also Inner product space
  More results at FactBites »

 
 

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