Why is division not a viable operation for vectors?

In summary, division is not possible between vectors because it requires a defined multiplication operation between vectors. While addition, subtraction, and multiplication between vectors are possible in some cases, it is not possible to define a consistent division operation due to the presence of "zero divisors" and the lack of a defined multiplication operation. This is in contrast to algebras, which do have a defined multiplication operation, and may be a closer concept to what is being discussed.
  • #1
abrowaqas
114
0
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
abrowaqas said:
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
Tac-Tics said:
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

Related to Why is division not a viable operation for vectors?

1. What is the definition of division between vectors?

The division between two vectors is a mathematical operation that results in a new vector. It is defined as the multiplication of the first vector by the reciprocal of the second vector.

2. How is division between vectors different from scalar division?

Division between vectors is different from scalar division because it results in a new vector, while scalar division results in a single number. In division between vectors, both magnitude and direction are taken into account.

3. Can you divide any two vectors?

No, division between vectors is only defined for two vectors in the same dimension. In other words, both vectors must have the same number of components or dimensions.

4. What is the result of dividing a vector by itself?

The result of dividing a vector by itself is a unit vector in the same direction as the original vector. This is because the reciprocal of a vector is the vector itself.

5. Are there any limitations to division between vectors?

Yes, there are some limitations to division between vectors. It is not defined for vectors with a magnitude of zero, as division by zero is undefined. Additionally, division between vectors is not commutative, meaning the order of the vectors matters.

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