mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...

– 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,

\mathbf \cdot \mathbf = , \mathbf,  \, , \mathbf,  \cos \theta

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 ...

– also known as the "vector product", a binary operation on two vectors that results in another

vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

. 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

\mathbf

is the unit vector perpendicular to the plane determined by vectors

\mathbf

and

\mathbf

\mathbf \times \mathbf = , \mathbf,  \, , \mathbf,  \sin \theta \, \mathbf.

Exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of ...

or wedge product – a binary operation on two vectors that results in a bivector. In Euclidean 3-space, the wedge product

\mathbf \wedge \mathbf

has the same magnitude as the cross product

\mathbf \times \mathbf

(the area of the parallelogram formed by sides

\mathbf

and

\mathbf

) but generalizes to arbitrary

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

s and products between more than two vectors. *

Tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...

– for two vectors

v\in V

and

w\in W,

where

V

and

W

are

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

s, their ''tensor product''

v\otimes w

belongs to the ''tensor product''

V\otimes W

of the vector spaces. * Geometric product or Clifford product – for two vectors, the geometric product

\mathbf \mathbf = \mathbf \cdot \mathbf + \mathbf \wedge \mathbf

is a mixed quantity consisting of a scalar plus a bivector. The geometric product is well defined for any

multivector In multilinear algebra, a multivector, sometimes called Clifford number or multor, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -ve ...

s as arguments. * A bilinear product in an

algebra over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...

. * A

Lie bracket 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 identit ...

for vectors 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 ident ...

. * Hadamard product – entrywise or elementwise product of

tuple In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...

s of scalar coordinates, where

(a \odot b)_i = a_i b_i

. *

Outer product In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions ''n'' and ''m'', the ...

- where

(\mathbf \otimes \mathbf)

with

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

results in a

(d \times d)

matrix. *

Triple product In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the ve ...

s – products involving three vectors. * Quadruple products – products involving four vectors.

Applications

Vector multiplication has multiple applications in regards to mathematics, but also in other studies such as physics and engineering.

Physics

* The

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 ...

occurs frequently in the study of

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

, where it is used to calculate

torque In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...

and

angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...

. It can also be used to calculate the

Lorentz force In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...

exerted on a charged particle moving in a magnetic field. * The

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

is used to determine the

work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an ani ...

done by a constant force.

Applications

Physics

See also