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Articles about Pseudovector |
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| CATEGORIES ABOUT PSEUDOVECTOR | |
| linear algebra | |
| vector calculus | |
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A common way of constructing a pseudovector p is by taking the Cross Product of two vectors '''a''' and '''b''': :p = '''a''' × '''b''' A simple example of an improper rotation in 3D (but not in 2D) is a coordinate inversion: x goes to −x, '''y''' to −'''y''' and '''z''' to −'''z'''. Under this transformation, '''a''' and '''b''' go to −'''a''' and −'''b''' (by the definition of a vector), but '''p''' clearly does not change. It follows that any improper rotation multiplies '''p''' by −1 compared to the rotation's effect on a true vector. This concept can be further generalized to Pseudoscalar s and '''pseudotensors''' , both of which gain an extra sign flip under improper rotations compared to a true Scalar or Tensor . Many occurrences of pseudovectors in mathematics and physics are more naturally analyzed as Bivector s, following the calculus of Differential Form s; the double negation is natural for a bivector. However, bivectors are "less intuitive" in some senses than ordinary vectors, and since in R3 every bivector '''a''' ∧ '''b''' has a unique Dual vector '''a''' × '''b''', it is this dual which is more often used. PHYSICAL EXAMPLES Physical examples of pseudovectors include the Magnetic Field , Torque , Vorticity , and the Angular Momentum . Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting (invariant) under mirror reflections through the plane (an improper rotation), but the magnetic field is anti-symmetric (flips sign) under that mirror plane—this contradiction is resolved by realizing that the mirror reflection of the field induces an extra sign flip because of its pseudovector nature. To the extent that physical laws are the same for right-handed and left-handed coordinate systems (i.e. invariant under inversion), the sum of a vector and a pseudovector is not meaningful. However, the Weak Force , which governs Beta Decay , ''does'' depend on the Chirality of the universe, and in this case pseudovectors and vectors ''are'' added. REFERENCES
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