In mathematics, the dot product or scalar product^{[note 1]} is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).

Algebraically, the dot product 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. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their lengths).

The name "dot product" is derived from the dot operator " · " that is often used to designate this operation;^[1] the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector (as with the vector product in three-dimensional space).

Definition

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

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

$${\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}}$$

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 ${\displaystyle }$ and ${\displaystyle }$ is:

$${\displaystyle {\begin{aligned}\ \cdot &=(1\times 4)+(3\times -2)+(-5\times -1)\\&=4-6+5\\&=3\end{aligned}}}$$

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

$${\displaystyle {\begin{aligned}\ 1,3,-5\cdot 1,3,-5&=(1\times 1)+(3\times 3)+(-5\times -5)\\&=1+9+25\\&=35\end{aligned}}}$$

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

$${\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\mathsf {T}}\mathbf {b} ,}$$

${\displaystyle a{^{\mathsf {T}}}}$

${\displaystyle \mathbf {a} }$

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:

$${\displaystyle {\begin{bmatrix}1&3&-5\end{bmatrix}}{\begin{bmatrix}4\\-2\\-1\end{bmatrix}}=3\,.}$$

Geometric definitionedit

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]

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

${\displaystyle \theta }$

${\displaystyle \mathbf {a} }$