Cross Product As An Exterior Product
The cross product can be viewed in terms of the exterior product. This view allows for a natural geometric interpretation of the cross product. In exterior algebra the exterior product (or wedge product) of two vectors is a bivector. A bivector is an oriented plane element, in much the same way that a vector is an oriented line element. Given two vectors a and b, one can view the bivector a ∧ b as the oriented parallelogram spanned by a and b. The cross product is then obtained by taking the Hodge dual of the bivector a ∧ b, mapping 2-vectors to vectors:
This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. Only in three dimensions is the result an oriented line element – a vector – whereas, for example, in 4 dimensions the Hodge dual of a bivector is two-dimensional – another oriented plane element. So, only in three dimensions is the cross product of a and b the vector dual to the bivector a ∧ b: it is perpendicular to the bivector, with orientation dependent on the coordinate system's handedness, and has the same magnitude relative to the unit normal vector as a ∧ b has relative to the unit bivector; precisely the properties described above.
Read more about this topic: Cross Product
Famous quotes containing the words cross, product and/or exterior:
“I’d take off all my clothes
& cross the damp cold lawn & down the bluff
into the terrible water & walk forever
under it out toward the island.”
—John Berryman (1914–1972)
“Whenever a taboo is broken, something good happens, something vitalizing.... Taboos after all are only hangovers, the product of diseased minds, you might say, of fearsome people who hadn’t the courage to live and who under the guise of morality and religion have imposed these things upon us.”
—Henry Miller (1891–1980)
“The competent leader of men cares little for the niceties of other peoples’ characters: he cares much—everything—for the exterior uses to which they may be put.... These are men to be moved. How should he move them? He supplies the power; others simply the materials on which that power operates.”
—Woodrow Wilson (1856–1924)