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Dot Product ○◂｜Definition｜1st｜20251119205401-00-⌔
Dot product
In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the scalar product1 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 (the length of a vector is the square root of the scalar product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their scalar 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;2 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).
Printed 2026-06-28.
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Link to original Footnotes
The term scalar product means literally “product with a scalar as a result”. It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused with scalar multiplication. ↩
“Dot Product”. www.mathsisfun.com. Retrieved 2020-09-06. ↩
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