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{{math-h}}
{{math-h}}
In the <math>n</math>-dimensional [[Euclidean vector space]] <math>\mathbb{R}^n</math>, the '''dot product''' is defined for two [[Vector|vectors]] <math>\vec{x} =
\langle x_1, \ldots, x_n \rangle</math> and <math>\vec{y} = \langle y_1, \ldots, y_n \rangle</math> as follows:
: <math>\vec{X} \cdot \vec{Y} = |\vec{X}||\vec{Y}|\cos\theta</math>
: <math>\vec{x} \cdot \vec{y} = \sum_{i=1}^n x_i y_i = x_1 y_1 + \cdots +x_n y_n </math>
Unlike the [[cross product]], the dot product is a [[scalar]], not a vector, and has no direction.
For example, in 3-dimensional Euclidean space, let <math>\vec{x} = \langle 1, 2, 3 \rangle</math> and <math>\vec{y} = \langle 4, 5, 6\rangle</math>.
Then we can calculate <math>\vec{x} \cdot \vec{y} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32</math>.
The dot product of any vector with itself is the square of its [[norm]].
In 2- and 3- dimensional Euclidean space, the relation between the dot product of two vectors and the angle between them is given by
: <math>\vec{x} \cdot \vec{y} = |\vec{x}||\vec{y}|\cos\theta</math>
where <math>|\cdot|</math> is the [[vector norm]] and <math>\theta</math> is the angle between the vectors. Note that the vectors are perpendicular
(or [[orthogonal]]) if and only if their dot product is zero.
This relationship can be extended to define the concept of the angle between vectors in higher-dimensional Euclidean vector spaces. Two vectors
in <math>\mathbb{R}^n</math> are defined to be [[orthogonal]] if their dot product is zero.
The dot product is the best-known example of an [[inner product]].
== Application ==
== Application ==