dot product

Also called scalar product, quantity product (scalar product). It is the sum of the products of corresponding entries of two sequences of numbers. 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 the projected product ) of a Euclidean space, and is a special case of an inner product, although it is not the only inner product that can be defined on a Euclidean space.

Algebraically, a dot product is the sum of the products of corresponding entries of two sequences of numbers. Geometrically, it is the product of the Euclidean magnitude of the two vectors and the cosine of the angle between them. These two definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are usually defined using vector spaces. In this case, the dot product is used to define the length (the length of a vector is the square root of the dot product of the vectors themselves) and the angle (the cosine of the angle between two vectors is equal to the quotient of their dot product and the product of their lengths).

algebra definition

geometric definition

relationship with inner product

cross product (corss product)

Or vector product ( vector product) (sometimes directed area product , to emphasize its geometric meaning) is in a three-dimensional directed Euclidean vector space, and denoted by the symbol x. Given two linearly independent vectors a and b , The cross product a × b (pronounced "a cross b") is a vector perpendicular to a and b , and therefore perpendicular to the plane containing them.

definition

The figure below shows the cross product of a and b using the Sarrus rule

The cross product can also be expressed as a determinant of the form:

This determinant can be calculated using Sarrus' rule or cofactor expansion. Using Sarrus' rule, it expands to

Following the first line and using cofactors instead, it expands to

It directly gives the weight of the result.

geometric meaning

inner product

The inner product of two vectors in a space is a scalar, usually denoted by angle brackets, eg. Inner product spaces generalize Euclidean vector spaces, where inner products are dot or scalar products of Cartesian coordinates.

Infinite-dimensional inner product spaces are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes called unitary spaces . The first use of the concept of a vector space with an inner product is due to Giuseppe Peano, 1898.

Geometric interpretation of the angle between two vectors defined using the inner product ( | x | and | y | are norms in 2D and 3D space)

definition

example

outer product

In linear algebra, the outer product of two coordinate vectors is a matrix. If these two vectors have dimensions n and m , then their outer product is an n × m matrix. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also known as tensor product and can be used to define tensor algebras.