Division between vectors

  • Thread starter abrowaqas
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  • #1
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Why division is not possible between vectors?
We have seen in many books that there is vector addition, subtraction and multiplication .. but division between two vectors or more vectors is not anywhere... what is the reason of not having division in vectors.does it have any connection with projections?
 

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  • #2
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Why division is not possible between vectors?
We have seen in many books that there is vector addition, subtraction and multiplication .. but division between two vectors or more vectors is not anywhere... what is the reason of not having division in vectors.does it have any connection with projections?

Multiplication is not possible between vectors either.

Algebras (over a field), on the other hand, are essentially vectorspaces with a multiplication on them.

You may also want to look at the difference between rings and fields. This is closer to what you're interested in.
 
  • #3
HallsofIvy
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Multiplication is not possible between vectors either.

Depends upon what definiton of "vector" you are using. In general Linear Algebra we have no "multiplication of vectors" but in some special vector spaces we do. For example, in the vector space of all polynomials, we can certainly define the product of "vectors". And in [itex]R^n[/itex], which is what I think the OP is talking about, we can define dot product. In [itex]R^3[/itex] specifically, we have the cross product of two vectors. That's the only product in which "division" might make sense because the dot product of two vectors is not a vector. And we know that the cross product of two parallel vectors is 0. That is, cross product has "zero divisors" so that "multiplicative inverses" are not defined for some non-zero vectors. And, therefore, we cannot define "division".

Algebras (over a field), on the other hand, are essentially vectorspaces with a multiplication on them.

You may also want to look at the difference between rings and fields. This is closer to what you're interested in.
Very good point!
 
  • #4
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