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tgt
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Why aren't there any legit operation for division of two vectors (any kind of vectors)?
Ask about multiplication first!tgt said:Why aren't there any legit operation for division of two vectors (any kind of vectors)?
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.
I was thinking of the algebra as a whole, rather than just the vectors.maze said:Can you please elaborate?
Hurkyl said:I was thinking of the algebra as a whole, rather than just the vectors.
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.
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.
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.
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.
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).
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.