Dot product
In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number. In Euclidean geometry, the scalar product of two vectors is the dot product of their Cartesian coordinates, and is independent from the choice of a particular Cartesian coordinate system. The terms "dot product" and "scalar product" are often used interchangeably when a Cartesian coordinate system has been fixed once for all. The scalar product being a particular inner product, the term "inner product" is also often used.
Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, the scalar product of two vectors is the product of their lengths and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the scalar product is used for defining lengths and angles.
The name "dot product" is derived from the dot operator " ⋅ " that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector.
Definition
The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance of vectors. The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space.In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space. In such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the angle between two vectors of length one is defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry.
Coordinate definition
The dot product of two vectors and specified with respect to an orthonormal basis, is defined as:where denotes summation and is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors and is:
Likewise, the dot product of the vector with itself is:
If vectors are identified with column vectors, the dot product can also be written as a matrix product
where denotes the transpose of.
Expressing the above example in this way, a 1 × 3 matrix is multiplied by a 3 × 1 matrix to get a 1 × 1 matrix that is identified with its unique entry:
Geometric definition
In Euclidean space, a Euclidean vector is a geometric 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 to which the arrow points. The magnitude of a vector is denoted by. The dot product of two Euclidean vectors and is defined bywhere is the angle between and.
In particular, if the vectors and are orthogonal, then, which implies that
At the other extreme, if they are codirectional, then the angle between them is zero with and
This implies that the dot product of a vector with itself is
which gives
the formula for the Euclidean length of the vector.
Scalar projection and first properties
The scalar projection of a Euclidean vector in the direction of a Euclidean vector is given bywhere is the angle between and.
In terms of the geometric definition of the dot product, this can be rewritten as
where is the unit vector in the direction of.
The dot product is thus characterized geometrically by
The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar,
It also satisfies the distributive law, meaning that
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 is never negative, and is zero if and only if, the zero vector.
Equivalence of the definitions
If are the standard basis vectors in, then we may writeThe vectors are an orthonormal basis, which means that they have unit length and are at right angles to each other. Since these vectors have unit length,
and since they form right angles with each other, if,
Thus in general, we can say that:
where is the Kronecker delta.
Also, by the geometric definition, for any vector and a vector, we note that
where is the component of vector in the direction of. The last step in the equality can be seen from the figure.
Now applying the distributivity of the geometric version of the dot product gives
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,, and are real vectors and,, and are scalars.; Commutative : which follows from the definition : The commutative property can also be easily proven with the algebraic definition, and in more general spaces the angle between two vectors can be defined as
; Bilinear :
; Not associative : Because the dot product is not defined between a scalar and a vector associativity is meaningless. However, bilinearity implies This property is sometimes called the "associative law for scalar and dot product", and one may say that "the dot product is associative with respect to scalar multiplication".
; Orthogonal : Two non-zero vectors and are orthogonal if and only if.
; No cancellation
; Product rule : If and are vector-valued differentiable functions, then the derivative of is given by the rule
Application to the law of cosines
Given two vectors and separated by angle , they form a triangle with a third side. Let, and denote the lengths of,, and, respectively. The dot product of with itself is:which is the law of cosines.
Triple product
There are two ternary operations involving dot product and cross product.The scalar triple product of three vectors is defined as
Its value is the determinant of the matrix whose columns are the Cartesian coordinates of the three vectors. It is the signed volume of the parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the exterior product of three vectors.
The vector triple product is defined by
This identity, also known as Lagrange's formula, may be remembered as "ACB minus ABC", keeping in mind which vectors are dotted together. This formula has applications in simplifying vector calculations in physics.
Physics
In physics, the dot product takes two vectors and returns a scalar quantity. It is also known as the "scalar product". The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Thus, Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector.For example:
- Mechanical work is the dot product of force and displacement vectors,
- Power is the dot product of force and velocity.
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 could be zero without the vector being the zero vector.Inner product
The inner product generalizes the dot product to abstract vector spaces over a field of scalars, being either the field of real numbers or the field of complex numbers. It is usually denoted using angular brackets by.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.
Functions
The dot product is defined for vectors that have a finite number of entries. Thus these vectors can be regarded as discrete functions: a length- vector is, then, a function with domain, and is a notation for the image of by the function/vector.This notion can be generalized to square-integrable functions: just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some measure space :
For example, if and are continuous functions over a compact subset of with the standard Lebesgue measure, the above definition becomes:
Generalized further to complex continuous functions and, by analogy with the complex inner product above, gives:
Weight function
Inner products can have a weight function. Explicitly, the inner product of functions and with respect to the weight function isDyadics and matrices
A double-dot product for matrices is the Frobenius inner product, which is analogous to the dot product on vectors. It is defined as the sum of the products of the corresponding components of two matrices and of the same size:And for real matrices,
Writing a matrix as a dyadic, we can define a different double-dot product however it is not an inner product.