Vector Division: Is It Possible?

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Discussion Overview

The discussion centers around the concept of vector division, exploring whether it is possible and under what conditions. Participants examine the mathematical foundations of vector operations, particularly focusing on multiplication and its implications for division, as well as the relevance of complex numbers and quaternions in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that traditional vector operations include addition, subtraction, and scalar multiplication, but questions the existence of a division operation for vectors.
  • Another participant explains that division requires a multiplication operation with a cancellation property and challenges the applicability of the cross product for this purpose.
  • A participant mentions that complex numbers provide a way to define multiplication in a two-dimensional vector space, which may allow for a form of division.
  • It is reiterated that division is fundamentally defined in terms of multiplication, with examples provided from real and complex number systems.
  • Concerns are raised about the implications of being a second-year physics undergrad in relation to understanding complex numbers, suggesting a potential disparity in educational backgrounds.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the possibility of vector division. While some explore the concept through the lens of complex numbers and quaternions, others assert that division is not generally defined for vectors.

Contextual Notes

The discussion highlights the dependence on definitions of multiplication and the mathematical structures involved, such as complex numbers and quaternions, without resolving the broader implications or limitations of vector division.

neutrino
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From school, we've had addition, "subtraction" of vectors and then scalar multiplication, dot & cross products. I have never come across dividing a vector by another. Is there such an operation?

Eagerly awaiting enlightening replies,
Nanveeth
 
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In order to have division one needs a multiplication which for which the cancellation property holds (amongst other things)

xy=xz => y=z

what multiplication do you want to use? cross product won't do...

actually there is a way of defining a multiplication such that there is a division for two dimensional vectors (have you heard of complex numbers?).
 
This is actually a general question that I have and I'm not planning to use it anywhere for now.

Btw, I'm a second-year physics undergrad.
 
And my answer still remains: division is defined in terms of multiplication. 1/2 is the number which when multiplied by 2 yields 1 etc.

The complex numbers are a two dimensional real vector space which posses such a multiplication. THe quaternions a 4-d real vector space.

In *general* we do not have a division of vectors.

I don't know what being a second year physics undergrad means per se in relation to my post (was it supposed to indicate you know what complex numbers are? If you were at Cambridge then you'd know what complex numbers are, if you were at some less than stellar state or private university in the US then you probably wouldn't)
 

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