Why it is not possible to have divisions of vectors?

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Discussion Overview

The discussion revolves around the question of why division of vectors is not possible, along with related inquiries about the vector product of polar and axial vectors. The scope includes conceptual understanding and technical reasoning regarding vector operations.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that the inability to divide vectors stems from the nature of vectors as functions that adhere to specific rules of addition, rather than being numbers that can be divided.
  • One participant notes that vectors are additive groups and implies that they do not form multiplicative groups, which may contribute to the understanding of why division is not defined.
  • There is uncertainty regarding the definitions of 'axial' and 'polar' vectors, with one participant requesting clarification on these terms.
  • Another participant expresses that division of vectors is not analogous to division of numbers, indicating a fundamental difference in their mathematical treatment.

Areas of Agreement / Disagreement

Participants express differing views on the nature of vectors and their operations, particularly regarding the definitions of axial and polar vectors. The discussion remains unresolved, with no consensus on the implications of these definitions or the nature of vector division.

Contextual Notes

Participants have not fully clarified their assumptions regarding the definitions of axial and polar vectors, which may affect the discussion. Additionally, the mathematical framework for vector operations is not fully explored, leaving some points open to interpretation.

wasi-uz-zaman
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hi,please answer some question about vectors?
a) why it is not possible to have divisions of vectors?
b) is vector product of two polar vector is always axial vector?
c)what is vector product of one axial vector and polar vector?
thanks
wasi-uz-zaman
 
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If you believe the "geometric" intuition of vectors then it is kinda obvious why, not? but if you dive a little deeper into the construction of a vector all it is a function which must obey these rules
## + : E \times E \to E##
##+(x_1,x_2) \mapsto x_1+x_2## and the axiom of a group with only this operation
[1]associative law of addition
[2]commutative law of addition
[3] inverse.
Also, you can only multiply scalars to vectors (cross product is a completely different operation which falls into something called the determinant function. this function has an inverse, if that is what you are asking.)
Could you clarify what you mean why axial vector and polar vector? as in rectangular and polar coordinates?

Edit: I mean that if you understand that Vectors are only additive groups then it should be helpful to understand that they aren't multiplicative groups.
 
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a) a vector's not a number like we think of numbers, so you can't divide with them (although there are plenty of other operations in higher-level math)

don't know about b or c though...
 
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It's not clear what the OP means by an 'axial' and a 'polar' vector.
 

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