Viktad skalärprodukt
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.
Skalärprodukt - Chalmers
It is often called the inner product or rarely the 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.
Skalarprodukt
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.
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 such a presentation, the notions of length and angle are defined by means of the dot product.
Dot product - Wikipedia
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.
If vectors are identified with column vectors , the dot product can also be written as a matrix product. 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. These properties may be summarized by saying that the dot product is a bilinear form.
The last step in the equality can be seen from the figure. So the geometric dot product equals the algebraic dot product. The dot product of this with itself is:. There are two ternary operations involving dot product and cross product. It is the signed volume of the parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the exterior product of three vectors.
This formula has applications in simplifying vector calculations in physics. In physics , the dot product takes two vectors and returns a scalar quantity. It is also known as the "scalar product". The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors.
Linjär algebra: är skalärprodukten den enda funktionen som kan vara en inre produkt i Rn?
For example: [ 10 ] [ 11 ]. 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.
This in turn would have consequences for notions like length and angle. When vectors are represented by column vectors , the dot product can be expressed as a matrix product involving a conjugate transpose , denoted with the superscript H:. 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.