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.
- 1 Definition
- 2 Properties
- 3 Triple product expansion
- 4 Physics
- 5 Generalizations
- 6 See also
- 7 References
- 8 External links
The dot product is often defined in one of two ways: algebraically or geometrically. The equivalence of these definitions is proven later.
The geometric definition is based on the notion of angle. It should be noted that, in the modern presentation of Euclidean geometry, the points are defined as coordinates vectors. In such a presentation of the geometry, the notions of length and angles are not primitive and need to be defined. Therefore, in this case, the length of a vector is defined as the square root of the dot product of the vector by itself, and the geometric definition of the dot product is inverted to define the notion of (non oriented) angle.
The dot product of two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is defined as:
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 . The dot product of two Euclidean vectors A and B is defined by
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
At the other extreme, if they are codirectional, then the angle between them is 0° and
This implies that the dot product of a vector A by itself is
the formula for the Euclidean length of the vector.
Scalar projection and the equivalence of the definitions
The scalar projection (or scalar component) of a Euclidean vector A in the direction of a Euclidean vector B is given by
where θ is the angle between A and B.
In terms of the geometric definition of the dot product, this can be rewritten
where is the unit vector in the direction of B.
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 α,
It also satisfies a distributive law, meaning that
As a consequence, if are the standard basis vectors in , then writing
which is precisely the algebraic definition of the dot product. More generally, the same identity holds with the ei replaced by any orthonormal basis.
- which follows from the definition (θ is the angle between a and b):
- Distributive over vector addition:
- Scalar multiplication:
- 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.
- Derivative: 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
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:
which is the law of cosines.
Triple product expansion
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.
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
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
The angle between two complex vectors is then given 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.
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 interval. For example, a the inner product of two real continuous functions u(x), v(x) may be defined on the interval a ≤ x ≤ b (also denoted [a, b]):
Inner products can have a weight function, i.e. a function which weight each term of the inner product with a value.
Dyadics and matrices
Matrices have the Frobenius inner product, which is analogous to the vector inner product. It is defined as the sum of the products of the corresponding components of two matrices A and B having the same size:
- (For real matrices)
- S. Lipschutz, M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1.
- M.R. Spiegel, S. Lipschutz, D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7.
- Arfken, G. B.; Weber, H. J. (2000). Mathematical Methods for Physicists (5th ed.). Boston, MA: Academic Press. pp. 14–15. ISBN 978-0-12-059825-0..
- K.F. Riley, M.P. Hobson, S.J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. ISBN 978-0-521-86153-3.
- M. Mansfield, C. O’Sullivan (2011). Understanding Physics (4th ed.). John Wiley & Sons. ISBN 978-0-47-0746370.
- Hazewinkel, Michiel, ed. (2001), "Inner product", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Dot product", MathWorld.
- Explanation of dot product including with complex vectors
- "Dot Product" by Bruce Torrence, Wolfram Demonstrations Project, 2007.