Therefore, for consistency, the other side must also be an axial vector. Namely, polar vectors and axial vectors are interrelated in the following ways under application of the cross product:.

Because the cross product may also be a polar vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a polar vector and the other one is an axial vector e. For instance, a vector triple product involving three polar vectors is a polar vector. Let i , j , k be an orthonormal basis.

The vectors i , j and k don't depend on the orientation of the space. They can even be defined in the absence of any orientation. They can not therefore be axial vectors. This is a paradox. The vectors i , j and k are mathematical vectors, neither axial nor polar. In mathematics, the cross-product of two vectors is a vector.

There is no contradiction. The cross product can be seen as one of the simplest Lie products, and is thus generalized by Lie algebras , which are axiomatized as binary products satisfying the axioms of multilinearity, skew-symmetry, and the Jacobi identity.

Many Lie algebras exist, and their study is a major field of mathematics, called Lie theory. For example, the Heisenberg algebra gives another Lie algebra structure on R 3. The cross product can also be described in terms of quaternions. The real part will be the negative of the dot product of the two vectors. A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions.

The nonexistence of nontrivial vector-valued cross products of two vectors in other dimensions is related to the result from Hurwitz's theorem that the only normed division algebras are the ones with dimension 1, 2, 4, and 8. In general dimension, there is no direct analogue of the binary cross product that yields specifically a vector.

There is however the exterior product, which has similar properties, except that the exterior product of two vectors is now a 2-vector instead of an ordinary vector. As mentioned above, the cross product can be interpreted as the exterior product in three dimensions by using the Hodge star operator to map 2-vectors to vectors. The exterior product and dot product can be combined through summation to form the geometric product in geometric algebra.

As mentioned above, the cross product can be interpreted in three dimensions as the Hodge dual of the exterior product. This generalization is called external product.

While the commutator product of two 1-vectors is indeed the same as the exterior product and yields a 2-vector, the commutator of a 1-vector and a 2-vector yields a true vector, corresponding instead to the left and right contractions in geometric algebra.

The commutator product of two 2-vectors has no corresponding equivalent product, which is why the commutator product is defined in the first place for 2-vectors. Furthermore, the commutator triple product of three 2-vectors is the same as the vector triple product of the same three pseudovectors in vector algebra. However, the commutator triple product of three 1-vectors in geometric algebra is instead the negative of the vector triple product of the same three true vectors in vector algebra.

Generalizations to higher dimensions is provided by the same commutator product of 2-vectors in higher-dimensional geometric algebras, but the 2-vectors are no longer pseudovectors. In the context of multilinear algebra , the cross product can be seen as the 1,2 -tensor a mixed tensor , specifically a bilinear map obtained from the 3-dimensional volume form , [note 2] a 0,3 -tensor, by raising an index.

The most direct generalizations of the cross product are to define either:. These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity. This formula is identical in structure to the determinant formula for the normal cross product in R 3 except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors v 1 , If n is odd, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product.

In the case that n is even, however, the distinction must be kept. And just like the vector cross product, it can be defined in a coordinate independent way as the Hodge dual of the wedge product of the arguments.

In , Joseph-Louis Lagrange used the component form of both the dot and cross products in order to study the tetrahedron in three dimensions. In , William Rowan Hamilton introduced the quaternion product, and with it the terms vector and scalar. James Clerk Maxwell used Hamilton's quaternion tools to develop his famous electromagnetism equations , and for this and other reasons quaternions for a time were an essential part of physics education.

In , Hermann Grassmann published a geometric algebra not tied to dimension two or three. Grassmann develops several products, including a cross product represented then by [uv]. In , Augustin-Louis Cauchy , a contemporary of Grassmann, published a paper on algebraic keys which were used to solve equations and had the same multiplication properties as the cross product.

In , William Kingdon Clifford published Elements of Dynamic , in which the term vector product is attested. In the book, this product of two vectors is defined to have magnitude equal to the area of the parallelogram of which they are two sides, and direction perpendicular to their plane.

Wilson keeps the term skew product , but observes that the alternative terms cross product [note 4] and vector product were more frequent.

From Wikipedia, the free encyclopedia. Mathematical operation on vectors in 3D space. This article is about the cross product of two vectors in three-dimensional Euclidean space. For other uses, see Cross product disambiguation. See also: Triple product.

Main article: Vector-valued function § Derivative and vector multiplication. Main article: Triple product. For other uses, see Xyzzy. See also: Lorentz force. This section possibly contains original research. Please improve it by verifying the claims made and adding inline citations. Statements consisting only of original research should be removed.

September Learn how and when to remove this template message. Main article: Lie algebra. Further information: quaternions and spatial rotation.

See also: Seven-dimensional cross product and Octonion. Main articles: Exterior algebra and Comparison of vector algebra and geometric algebra § Cross and exterior products. Main articles: Geometric algebra § Extensions of the inner and exterior products , Cross product § Cross product and handedness , and Cross product § Lie algebra. Retrieved December The American Mathematical Monthly. doi : JSTOR S2CID Archived from the original PDF on If one requires only three basic properties of the cross product it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space.

Mathematical Methods for Physicists 4th ed. Methods of mathematical physics. Cambridge University Press. OCLC Elementary Fluid Dynamics. Oxford University Press. ISBN Practical Applied Mathematics. Zill; Michael R. Cullen Advanced engineering mathematics 3rd ed. ISBN X. Direct and Skew Products of Vectors". Vector Analysis. Founded upon the lectures of J. William Gibbs. New Haven: Yale University Press.

The dot product is called "direct product", and cross product is called "skew product". Crowe, Math. UC Davis. cited work. Spiegel; S. Lipschutz; D. Spellman Schaum's outlines. McGraw Hill. The American Mathematical Monthly, Vol.

Boichenko; Gennadiĭ Alekseevich Leonov; Volker Reitmann Dimension theory for ordinary differential equations. Clifford algebras and spinors 2nd ed.

Int J Information and Systems Sciences. Institute for scientific computing and education. Weisstein CRC concise encyclopedia of mathematics 2nd ed.

CRC Press. Clifford algebras and spinors. Cambridge: Cambridge University Press. Multilinear Algebra. Handbook of Linear Algebra. Understanding Geometric Algebra for Electromagnetic Theory.

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Navigation menu Personal tools Not logged in Talk Contributions Create account Log in. Learn more. When you think of leaders in the computing industry, your first thoughts probably turn to companies like Apple, Microsoft, and IBM. DEC identified a demand for more affordable and high-performing computing systems that could be used in scientific research and other technological settings. DEC introduced a mid-range computing solution, the minicomputer, at a time when the alternative was too bulky and costly for most people.

In addition to its extensive lines of minicomputers, it also became involved with software as well as the internet in the very, very early days of the internet. So, what happened to DEC? Why did all the success and industry dominance they had for roughly three decades just suddenly slip away? It ends in roughly with the selling of the company to Compaq. While working together in the Lincoln Laboratory at MIT, Ken Olsen and Harlan Anderson came up with the idea for DEC.

Olsen and Anderson also knew that these interactive machines could be made and sold cheaper, which would be another major draw for researchers and scientists who were craving the technology. In sticking with the theme of keeping expenses light not only for customers but for the company as well , they set up shop in an old wool mill in Maynard, MA. The first product DEC created and sold was its Digital Laboratory Module. Staying true to its business plan, the initial product lines DEC focused on were modules, or electronic components, that were mounted to circuit boards.

DEC began selling its first computer at the end of While continuing to release new PDPs into the market, DEC also charged forward in its delivery of new modules. The Flip Chip came out in and was meant to convert the PDP-4 to the PDP Many of its subsequent module releases served a similar purpose: helping users convert their old computers to upgraded versions. It was in this year when DEC released the PDP-8, which is widely recognized as the first successful commercial minicomputer.

In the interim, DEC came up with a revamped version of their PDP line and released the PDP minicomputer. Not only did it bring major upgraded features to their computing machines, it also was easier to use.

By the time it stopped selling it in the s, DEC sold over , of them, making it one of the most popular minicomputers ever. In addition, the design of the computer, as well as its operating system, turned out to be immensely popular with other computing companies, that eventually ended up using it as inspiration for their own work. After widespread success with its PDP, DEC made the move into high-end computers and launched the Virtual Address eXtension, or VAX. This new bit minicomputer or supermini line aimed to provide users with a wide array of computing resources that would be more affordable, powerful, and smaller than what companies like IBM could offer at the time.

DEC continued to stay busy during this time, regularly putting out new models of the VAX.

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