In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles: *

Dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...

– also known as the "scalar product", a binary operation that takes two vectors and returns a scalar quantity. 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. Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector. Thus, *::A ⋅ B = , A, , B, cos θ ** More generally, a bilinear product in an algebra over a field. *

Cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...

– also known as the "vector product", a binary operation on two vectors that results in another vector. The cross product of two vectors in 3-space is defined as the vector perpendicular to the plane determined by the two vectors whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle between the two vectors. So, if n̂ is the unit vector perpendicular to the plane determined by vectors A and B, *::A × B = , A, , B, sin θ n̂ ** More generally, a Lie bracket in a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

. * Hadamard product – entrywise product of vectors, where

(A \odot B)_i = A_i B_i

. *

Outer product In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of n ...

- where

(\mathbf \otimes \mathbf)

with

\mathbf \in \mathbb^d, \mathbf \in \mathbb^d

results in a

(d \times d)

matrix. * Triple products – products involving three vectors. * Multiple cross products – products involving more than three vectors.

See also