The dot product may be defined 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.

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 ${\displaystyle \mathbf {R} ^{n}}$ . 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 (non oriented) 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 edit

The dot product of two vectors ${\displaystyle \mathbf {a} =[a_{1},a_{2},\cdots ,a_{n}]}$  and ${\displaystyle \mathbf {b} =[b_{1},b_{2},\cdots ,b_{n}]}$ , specified with respect to an orthonormal basis, is defined as:^[2]

where ${\displaystyle \Sigma }$  denotes summation and ${\displaystyle n}$  is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors ${\displaystyle [1,3,-5]}$  and ${\displaystyle [4,-2,-1]}$  is:

Likewise, the dot product of the vector ${\displaystyle [1,3,-5]}$  with itself is:

If vectors are identified with column vectors, the dot product can also be written as a matrix product

Expressing the above example in this way, a 1 × 3 matrix (row vector) is multiplied by a 3 × 1 matrix (column vector) to get a 1 × 1 matrix that is identified with its unique entry:

Geometric definition edit

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 ${\displaystyle \mathbf {a} }$  is denoted by ${\displaystyle \left\|\mathbf {a} \right\|}$ . The dot product of two Euclidean vectors ${\displaystyle \mathbf {a} }$  and ${\displaystyle \mathbf {b} }$  is defined by^[3][4][1]

In particular, if the vectors ${\displaystyle \mathbf {a} }$  and ${\displaystyle \mathbf {b} }$  are orthogonal (i.e., their angle is ${\displaystyle {\frac {\pi }{2}}}$  or ${\displaystyle 90^{\circ }}$ ), then ${\displaystyle \cos {\frac {\pi }{2}}=0}$ , which implies that

Scalar projection and first properties edit

The scalar projection (or scalar component) of a Euclidean vector ${\displaystyle \mathbf {a} }$  in the direction of a Euclidean vector ${\displaystyle \mathbf {b} }$  is given by

In terms of the geometric definition of the dot product, this can be rewritten as

The dot product is thus characterized geometrically by^[5]

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 ${\displaystyle \mathbf {a} \cdot \mathbf {a} }$  is never negative, and is zero if and only if ${\displaystyle \mathbf {a} =\mathbf {0} }$ , the zero vector.

Equivalence of the definitions edit

If ${\displaystyle \mathbf {e} _{1},\cdots ,\mathbf {e} _{n}}$  are the standard basis vectors in ${\displaystyle \mathbf {R} ^{n}}$ , then we may write

Also, by the geometric definition, for any vector ${\displaystyle \mathbf {e} _{i}}$  and a vector ${\displaystyle \mathbf {a} }$ , we note that

Now applying the distributivity of the geometric version of the dot product gives

The dot product fulfills the following properties if ${\displaystyle \mathbf {a} }$ , ${\displaystyle \mathbf {b} }$ , and ${\displaystyle \mathbf {c} }$  are real vectors and ${\displaystyle r}$ , ${\displaystyle c_{1}}$  and ${\displaystyle c_{2}}$  are scalars.^[2][3]

Commutative

$${\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {b} \cdot \mathbf {a} ,}$$

 

which follows from the definition (${\displaystyle \theta }$  is the angle between ${\displaystyle \mathbf {a} }$  and ${\displaystyle \mathbf {b} }$ ):^[6]

$${\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta =\left\|\mathbf {b} \right\|\left\|\mathbf {a} \right\|\cos \theta =\mathbf {b} \cdot \mathbf {a} .}$$

 

Distributive over vector addition

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

 

Bilinear

$${\displaystyle \mathbf {a} \cdot (r\mathbf {b} +\mathbf {c} )=r(\mathbf {a} \cdot \mathbf {b} )+(\mathbf {a} \cdot \mathbf {c} ).}$$

 

Scalar multiplication

$${\displaystyle (c_{1}\mathbf {a} )\cdot (c_{2}\mathbf {b} )=c_{1}c_{2}(\mathbf {a} \cdot \mathbf {b} ).}$$

 

Not associative

because the dot product between a scalar ${\displaystyle \mathbf {a} \cdot \mathbf {b} }$  and a vector ${\displaystyle \mathbf {c} }$  is not defined, which means that the expressions involved in the associative property, ${\displaystyle (\mathbf {a} \cdot \mathbf {b} )\cdot \mathbf {c} }$  or ${\displaystyle \mathbf {a} \cdot (\mathbf {b} \cdot \mathbf {c} )}$  are both ill-defined.^[7] Note however that the previously mentioned scalar multiplication property is sometimes called the "associative law for scalar and dot product"^[8] or one can say that "the dot product is associative with respect to scalar multiplication" because ${\displaystyle c(\mathbf {a} \cdot \mathbf {b} )=(c\mathbf {a} )\cdot \mathbf {b} =\mathbf {a} \cdot (c\mathbf {b} )}$ .^[9]

Orthogonal

Two non-zero vectors ${\displaystyle \mathbf {a} }$  and ${\displaystyle \mathbf {b} }$  are orthogonal if and only if ${\displaystyle \mathbf {a} \cdot \mathbf {b} =0}$ .

No cancellation

Unlike multiplication of ordinary numbers, where if ${\displaystyle ab=ac}$ , then ${\displaystyle b}$  always equals ${\displaystyle c}$  unless ${\displaystyle a}$  is zero, the dot product does not obey the cancellation law:

If ${\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} }$  and ${\displaystyle \mathbf {a} \neq \mathbf {0} }$ , then we can write: ${\displaystyle \mathbf {a} \cdot (\mathbf {b} -\mathbf {c} )=0}$  by the distributive law; the result above says this just means that ${\displaystyle \mathbf {a} }$  is perpendicular to ${\displaystyle (\mathbf {b} -\mathbf {c} )}$ , which still allows ${\displaystyle (\mathbf {b} -\mathbf {c} )\neq \mathbf {0} }$ , and therefore allows ${\displaystyle \mathbf {b} \neq \mathbf {c} }$ .

Product rule

If ${\displaystyle \mathbf {a} }$  and ${\displaystyle \mathbf {b} }$  are vector-valued differentiable functions, then the derivative (denoted by a prime ${\displaystyle {}'}$ ) of ${\displaystyle \mathbf {a} \cdot \mathbf {b} }$  is given by the rule

$${\displaystyle (\mathbf {a} \cdot \mathbf {b} )'=\mathbf {a} '\cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {b} '.}$$

 

Application to the law of cosines edit

Given two vectors ${\displaystyle {\color {red}\mathbf {a} }}$  and ${\displaystyle {\color {blue}\mathbf {b} }}$  separated by angle ${\displaystyle \theta }$  (see image right), they form a triangle with a third side ${\displaystyle {\color {orange}\mathbf {c} }={\color {red}\mathbf {a} }-{\color {blue}\mathbf {b} }}$ . Let ${\displaystyle a}$ , ${\displaystyle b}$  and ${\displaystyle c}$  denote the lengths of ${\displaystyle {\color {red}\mathbf {a} }}$ , ${\displaystyle {\color {blue}\mathbf {b} }}$ , and ${\displaystyle {\color {orange}\mathbf {c} }}$ , respectively. The dot product of this with itself is:

which is the law of cosines.

There are two ternary operations involving dot product and cross product.

The scalar triple product of three vectors is defined as

The vector triple product is defined by^[2][3]

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. For example:^[10][11]

• Mechanical work is the dot product of force and displacement vectors,
• Power is the dot product of force and velocity.

Complex vectors edit

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 (e.g. this would happen with the vector ${\displaystyle \mathbf {a} =[1\ i]}$ ). 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 dot product, through the alternative definition^[12][2]

In the case of vectors with real components, this definition is the same as in the real case. The dot product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, the complex dot product is sesquilinear rather than bilinear, as it is conjugate linear and not linear in ${\displaystyle \mathbf {a} }$ . The dot product is not symmetric, since

The complex dot product leads to the notions of Hermitian forms and general inner product spaces, which are widely used in mathematics and physics.

The self dot product of a complex vector ${\displaystyle \mathbf {a} \cdot \mathbf {a} =\mathbf {a} ^{\mathsf {H}}\mathbf {a} }$ , involving the conjugate transpose of a row vector, is also known as the norm squared, ${\textstyle \mathbf {a} \cdot \mathbf {a} =\|\mathbf {a} \|^{2}}$ , after the Euclidean norm; it is a vector generalization of the absolute square of a complex scalar (see also: squared Euclidean distance).

Inner product edit

The inner product generalizes the dot product to abstract vector spaces over a field of scalars, being either the field of real numbers ${\displaystyle \mathbb {R} }$  or the field of complex numbers ${\displaystyle \mathbb {C} }$ . It is usually denoted using angular brackets by ${\displaystyle \left\langle \mathbf {a} \,,\mathbf {b} \right\rangle }$ .

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.

Functions edit

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-${\displaystyle n}$  vector ${\displaystyle u}$  is, then, a function with domain ${\displaystyle \{k\in \mathbb {N} :1\leq k\leq n\}}$ , and ${\displaystyle u_{i}}$  is a notation for the image of ${\displaystyle i}$  by the function/vector ${\displaystyle u}$ .

This notion can be generalized to continuous 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 interval [a, b]:^[2]

Generalized further to complex functions ${\displaystyle \psi (x)}$  and ${\displaystyle \chi (x)}$ , by analogy with the complex inner product above, gives^[2]

Weight function edit

Inner products can have a weight function (i.e., a function which weights each term of the inner product with a value). Explicitly, the inner product of functions ${\displaystyle u(x)}$  and ${\displaystyle v(x)}$  with respect to the weight function ${\displaystyle r(x)>0}$  is

Dyadics and matrices edit

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 ${\displaystyle \mathbf {A} }$  and ${\displaystyle \mathbf {B} }$  of the same size:

Writing a matrix as a dyadic, we can define a different double-dot product (see Dyadics § Product of dyadic and dyadic) however it is not an inner product.

Tensors edit

The inner product between a tensor of order ${\displaystyle n}$  and a tensor of order ${\displaystyle m}$  is a tensor of order ${\displaystyle n+m-2}$ , see Tensor contraction for details.

Algorithms edit

The straightforward algorithm for calculating a floating-point dot product of vectors can suffer from catastrophic cancellation. To avoid this, approaches such as the Kahan summation algorithm are used.

Libraries edit

A dot product function is included in:

• BLAS level 1 real SDOT, DDOT; complex CDOTU, ZDOTU = X^T * Y, CDOTC, ZDOTC = X^H * Y
• Fortran as dot_product(A,B) or sum(conjg(A) * B)
• Julia as  A' * B or standard library LinearAlgebra as dot(A, B)
• Matlab as  A' * B  or  conj(transpose(A)) * B  or  sum(conj(A) .* B)  or  dot(A, B)
• Python (package NumPy) as  numpy.dot(A, B)  or  numpy.inner(A, B)
• GNU Octave as  sum(conj(X) .* Y, dim), and similar code as Matlab
• Intel oneAPI Math Kernel Library real p?dot dot = sub(x)'*sub(y); complex p?dotc dotc = conjg(sub(x)')*sub(y)

• "Inner product", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Explanation of dot product including with complex vectors
• "Dot Product" by Bruce Torrence, Wolfram Demonstrations Project, 2007.