Dot product

"Scalar product" redirects here. For the abstract scalar product, see Inner product space. For the product of a vector and a scalar, see Scalar multiplication.
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 Euclidean 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.

Contents

  • 1 Definition
    • 1.1 Algebraic definition
    • 1.2 Geometric definition
    • 1.3 Scalar projection and first properties
    • 1.4 Equivalence of the definitions
  • 2 Properties
    • 2.1 Application to the cosine law
  • 3 Triple product expansion
  • 4 Physics
  • 5 Generalizations
    • 5.1 Complex vectors
    • 5.2 Inner product
    • 5.3 Functions
    • 5.4 Weight function
    • 5.5 Dyadics and matrices
    • 5.6 Tensors

      Definition

      The dot product is often defined in one of two ways: algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space.

      Algebraic definition

      The dot product of two vectors A = [A1, A2, ..., An] and B = [B1, B2, ..., Bn] is defined as:
      \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 and n is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors [1, 3, −5] and [4, −2, −1] is:
      \begin{align} \ [1, 3, -5] \cdot [4, -2, -1] &= (1)(4) + (3)(-2) + (-5)(-1) \\ &= 4 - 6 + 5 \\ &= 3. \end{align}

      Geometric definition

      In Euclidean space, a Euclidean vector is a geometrical object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by \|\mathbf{A}\|. The dot product of two Euclidean vectors A and B is defined by
      \mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|\cos\theta,
      where θ is the angle between A and B.
      In particular, if A and B are orthogonal, then the angle between them is 90° and
      \mathbf A\cdot\mathbf B=0.
      At the other extreme, if they are codirectional, then the angle between them is 0° and
      \mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|
      This implies that the dot product of a vector A by itself is
      \mathbf A\cdot\mathbf A = \|\mathbf A\|^2,
      which gives
      \|\mathbf A\| = \sqrt{\mathbf A\cdot\mathbf A},
      the formula for the Euclidean length of the vector.

      Scalar projection and first properties

      Scalar projection
      The scalar projection (or scalar component) of a Euclidean vector A in the direction of a Euclidean vector B is given by
      A_B=\|\mathbf A\|\cos\theta
      where θ is the angle between A and B.
      In terms of the geometric definition of the dot product, this can be rewritten
      A_B = \mathbf A\cdot\widehat{\mathbf B}
      where \widehat{\mathbf B} = \mathbf B/\|\mathbf B\| is the unit vector in the direction of B.
      Distributive law for the dot product
      The dot product is thus characterized geometrically by
      \mathbf A\cdot\mathbf B = A_B\|\mathbf{B}\|=B_A\|\mathbf{A}\|.
      The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar α,
      (\alpha\mathbf{A})\cdot\mathbf B=\alpha(\mathbf A\cdot\mathbf B)=\mathbf A\cdot(\alpha\mathbf B).
      It also satisfies a distributive law, meaning that
      \mathbf A\cdot(\mathbf B+\mathbf C) = \mathbf A\cdot\mathbf B+\mathbf A\cdot\mathbf C.
      These properties may be summarized by saying that the dot product is a bilinear form. Moreover, this bilinear form is positive definite, which means that \mathbf A\cdot \mathbf A is never negative and is zero if and only if \mathbf A = \mathbf 0.

      Equivalence of the definitions

      If e1,...,en are the standard basis vectors in Rn, then we may write
      \begin{align} \mathbf A &= [A_1,\dots,A_n] = \sum_i A_i\mathbf e_i\\ \mathbf B &= [B_1,\dots,B_n] = \sum_i B_i\mathbf e_i. \end{align}
      The vectors ei are an orthonormal basis, which means that they have unit length and are at right angles to each other. Hence since these vectors have unit length
      \mathbf e_i\cdot\mathbf e_i=1
      and since they form right angles with each other, if i ≠ j,
      \mathbf e_i\cdot\mathbf e_j = 0.
      Also, by the geometric definition, for any vector ei and a vector A, we note
      \mathbf A\cdot\mathbf e_i = \|\mathbf A\|\,\|\mathbf e_i\|\cos\theta = \|\mathbf A\|\cos\theta = A_i,
      where Ai is the component of vector A in the direction of ei.
      Now applying the distributivity of the geometric version of the dot product gives
      \mathbf A\cdot\mathbf B = \mathbf A\cdot\sum_i B_i\mathbf e_i = \sum_i B_i(\mathbf A\cdot\mathbf e_i) = \sum_i B_iA_i
      which is precisely the algebraic definition of the dot product. So the (geometric) dot product equals the (algebraic) dot product.

      Properties

      The dot product fulfills the following properties if a, b, and c are real vectors and r is a scalar.
    • Commutative:
      \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}.
      which follows from the definition (θ is the angle between a and b):
      \mathbf{a}\cdot \mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta = \|\mathbf{b}\|\|\mathbf{a}\|\cos\theta = \mathbf{b}\cdot\mathbf{a}
    • Distributive over vector addition:
      \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}.
    • Bilinear:
      \mathbf{a} \cdot (r\mathbf{b} + \mathbf{c}) = r(\mathbf{a} \cdot \mathbf{b}) + (\mathbf{a} \cdot \mathbf{c}).
    • Scalar multiplication:
      (c_1\mathbf{a}) \cdot (c_2\mathbf{b}) = c_1 c_2 (\mathbf{a} \cdot \mathbf{b})
    • Orthogonal:
      Two non-zero vectors a and b are orthogonal if and only if a ⋅ b = 0.
    • No cancellation:
      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:
      If a ⋅ b = a ⋅ c and a ≠ 0, then we can write: a ⋅ (b − c) = 0 by the distributive law; the result above says this just means that a is perpendicular to (b − c), which still allows (b − c) ≠ 0, and therefore b ≠ c.
    • Product Rule: If a and b are functions, then the derivative (denoted by a prime ′) of a ⋅ b is a′ ⋅ b + a ⋅ b′.

    Application to the cosine law

    Triangle with vector edges a and b, separated by angle θ.
    Main article: law of cosines
    Given two vectors a and b separated by angle θ (see image right), they form a triangle with a third side c = a − b. The dot product of this with itself is:
    \begin{align} \mathbf{c}\cdot\mathbf{c} & = (\mathbf{a}-\mathbf{b})\cdot(\mathbf{a}-\mathbf{b}) \\ & =\mathbf{a}\cdot\mathbf{a} - \mathbf{a}\cdot\mathbf{b} - \mathbf{b}\cdot\mathbf{a} + \mathbf{b}\cdot\mathbf{b}\\ & = a^2 - \mathbf{a}\cdot\mathbf{b} - \mathbf{a}\cdot\mathbf{b} + b^2\\ & = a^2 - 2\mathbf{a}\cdot\mathbf{b} + b^2\\ c^2 & = a^2 + b^2 - 2ab\cos \theta\\ \end{align}
    which is the law of cosines.

    Triple product expansion

    Main article: Triple product
    This is an identity (also known as Lagrange's formula) involving the dot- and cross-products. It is written as:
    \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})
    which may be remembered as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula finds application in simplifying vector calculations in physics.

    Physics

    In physics, vector 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. Examples include:
  • Mechanical work is the dot product of force and displacement vectors.
  • Magnetic flux is the dot product of the magnetic field and the area vectors.

Generalizations

Complex vectors

For vectors with complex entries, using the given definition of the dot product would lead to quite different properties. For instance the dot product of a vector with itself would be an arbitrary complex number, and could be zero without the vector being the zero vector (such vectors are called isotropic); this in turn would have consequences for notions like length and angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up the symmetric and bilinear properties of the scalar product, through the alternative definition.
\mathbf{a}\cdot \mathbf{b} = \sum{a_i \overline{b_i}}
where bi is the complex conjugate of bi. Then the scalar product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However this scalar product is thus sesquilinear rather than bilinear: it is conjugate linear and not linear in b, and the scalar product is not symmetric, since
\mathbf{a} \cdot \mathbf{b} = \overline{\mathbf{b} \cdot \mathbf{a}}.
The angle between two complex vectors is then given by
\cos\theta = \frac{\operatorname{Re}(\mathbf{a}\cdot\mathbf{b})}{\|\mathbf{a}\|\,\|\mathbf{b}\|}.
This type of scalar product is nevertheless useful, and leads to the notions of Hermitian form and of general inner product spaces.

Inner product

Main article: Inner product space
The inner product generalizes the dot product to abstract vector spaces over a field of scalars, being either the field of real numbers \mathbb{R} or the field of complex numbers \mathbb{C}. It is usually denoted by \langle\mathbf{a}\, , \mathbf{b}\rangle.
The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is sesquilinear instead of bilinear. An inner product space is a normed vector space, and the inner product of a vector with itself is real and positive-definite.
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