Division of Vectors: Legit Operations Explained

In summary, In geometric algebra, it is possible to divide vectors by interpreting the result as a geometric object that satisfies a specific equation. However, geometric algebra has many zero-divisors and is noncommutative, making it difficult to define a coherent division operation. Additionally, in order for division to make sense, there must be a well-defined way to multiply vectors.
  • #1
tgt
522
2
Why aren't there any legit operation for division of two vectors (any kind of vectors)?
 
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  • #2
In a geometric algebra it makes sense to divide vectors. If a and b are vectors, then the result of a/b would be interpreted as the geometric object such that (a/b)*b = a, where * is the geometric product.
 
  • #3
what is a geometric product? It is a rotation, reflection or translation?
 
  • #5
tgt said:
Why aren't there any legit operation for division of two vectors (any kind of vectors)?
Ask about multiplication first!


I'm skeptical about maze's comment, because geometric algebra has too many zero-divisors; there are generally lots of solutions to equations like bx = a. Furthermore, it's noncommutative, so a solution to bx=a might not be a solution to xb=a. I expect it to be hard to define any sort of coherent division operation.
 
  • #6
Hurkyl said:
Ask about multiplication first!


I'm skeptical about maze's comment, because geometric algebra has too many zero-divisors; there are generally lots of solutions to equations like bx = a. Furthermore, it's noncommutative, so a solution to bx=a might not be a solution to xb=a. I expect it to be hard to define any sort of coherent division operation.

Can you please elaborate? The solution to b*x = a is x=b-1*a where b-1=b/||b||. The solution to x*b = a is x = a*b-1 = 1/||b||(a.b+a^b) = 1/||b||(b.a-b^a) so it is almost the same as the solution to b*x = a, except the 2-form portion has the opposite sign.
 
  • #7
maze said:
Can you please elaborate?
I was thinking of the algebra as a whole, rather than just the vectors. :frown:

Specific example: if v is a unit vector, then (1+v)(1-v) = 0, so neither 1+v nor 1-v are invertible in the geometric algebra.

For the opening poster: if you're just working in the vector space, expressions like 1+v are nonsense. They only have meaning in a structure that supports such an operation, like a geometric algebra.
 
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  • #8
Well, this thread got a bit sidetracked. :)

Before it makes sense to talk about division of vectors, it better make sense to talk about multiplication of vectors. (The geometric algebra that was being discussed talks about one such way.) The natural way to define division from multiplication is this: if c = ab (where a, b, and c are just some abstract things) and you can find an inverse of, say, b (call it b-1), then you should have a = cb-1. (You might write a = c/b.)

You're aware, I hope, of two ways to combine two vectors (in [itex]\mathbb{R}^3[/itex]): the dot product and the cross product. The dot product takes two vectors and gives you a scalar, so that won't help you in trying to divide two vectors. The other one, the cross product, does give you a vector from two vectors, but the main issue is that if you are given vectors c and b and are told that c = a × b, then there isn't a unique solution for a. (For example, given 0 = a × b, you could take a = kb, where k is any scalar.)

Pretty much what this amounts to is this: in order for it to make sense to divide vectors, you probably need to find another way to multiply them first.
 
  • #9
I should point out that a useful product doesn't have to be of two vectors. e.g. he should know of the product
{scalar} * {vector} = {vector}

For some pairs of vectors, one can solve for a scalar x that satisfies xv=w (and the solution is unique). Of course, most pairs of vectors do not admit a quotient in this manner.
 
  • #10
Hurkyl said:
I was thinking of the algebra as a whole, rather than just the vectors. :frown:

Specific example: if v is a unit vector, then (1+v)(1-v) = 0, so neither 1+v nor 1-v are invertible in the geometric algebra.

For the opening poster: if you're just working in the vector space, expressions like 1+v are nonsense. They only have meaning in a structure that supports such an operation, like a geometric algebra.

ahh thanks for the explanation. For vectors though there are no such issues.
 

1. What is the purpose of dividing vectors?

The purpose of dividing vectors is to break down a single vector into multiple smaller vectors, known as components. This allows for a more detailed analysis of the vector's direction and magnitude.

2. How do you divide vectors mathematically?

To divide vectors, you must first determine the components of the original vector. Then, using the given mathematical operations, divide each component by the same number to obtain the new components of the divided vector.

3. Can you divide a vector by a scalar?

Yes, a vector can be divided by a scalar. This results in the magnitude of the vector being divided by the scalar, while the direction remains the same.

4. What is the difference between scalar division and vector division?

Scalar division involves dividing a single number by another number, while vector division involves dividing a vector into smaller components. Scalar division results in a single number, while vector division results in multiple numbers (the components).

5. Are there any special rules for dividing vectors?

Yes, there are a few special rules for dividing vectors. One of these rules is that the direction of the divided vector will be the same as the original vector, regardless of the mathematical operations used. In addition, dividing a vector by its magnitude will result in a unit vector in the same direction as the original vector.

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