Why it is not possible to have divisions of vectors?

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In summary, the conversation discusses the limitations of dividing vectors and the difference between vector and cross products. It also raises questions about the terms "axial vector" and "polar vector." The expert suggests that understanding vectors as additive groups can clarify their properties.
  • #1
wasi-uz-zaman
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hi,please answer some question about vectors?
a) why it is not possible to have divisions of vectors?
b) is vector product of two polar vector is always axial vector?
c)what is vector product of one axial vector and polar vector?
thanks
wasi-uz-zaman
 
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  • #2
If you believe the "geometric" intuition of vectors then it is kinda obvious why, not? but if you dive a little deeper into the construction of a vector all it is a function which must obey these rules
## + : E \times E \to E##
##+(x_1,x_2) \mapsto x_1+x_2## and the axiom of a group with only this operation
[1]associative law of addition
[2]commutative law of addition
[3] inverse.
Also, you can only multiply scalars to vectors (cross product is a completely different operation which falls into something called the determinant function. this function has an inverse, if that is what you are asking.)
Could you clarify what you mean why axial vector and polar vector? as in rectangular and polar coordinates?

Edit: I mean that if you understand that Vectors are only additive groups then it should be helpful to understand that they aren't multiplicative groups.
 
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  • #3
a) a vector's not a number like we think of numbers, so you can't divide with them (although there are plenty of other operations in higher-level math)

don't know about b or c though...
 
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  • #4
It's not clear what the OP means by an 'axial' and a 'polar' vector.
 
  • #5


a) It is not possible to divide vectors because division is not defined for vectors. Vectors represent magnitude and direction, not numerical values that can be divided. Division is only defined for scalars (numbers).
b) No, the vector product of two polar vectors can result in either a polar or axial vector, depending on the direction of the resulting vector.
c) The vector product of one axial vector and one polar vector will always result in an axial vector. This is because the cross product of two vectors always results in a vector perpendicular to both input vectors, and since axial vectors are perpendicular to polar vectors, the resulting vector must also be axial.
 

1. Why can't we divide vectors?

Dividing vectors is not possible because vectors are not simply numbers, but they also have a direction and magnitude. Dividing two vectors would require finding a direction and magnitude for the resulting vector, which is not a well-defined operation.

2. Can't we just divide the individual components of the vectors?

No, dividing the individual components of vectors does not give us a meaningful result. This is because the components of a vector are dependent on each other, and dividing them would not accurately represent the original vector.

3. What about dividing a vector by a scalar?

Dividing a vector by a scalar is possible, but it does not result in a new vector. Instead, it changes the magnitude of the original vector. This concept is known as scaling a vector.

4. Are there any exceptions where dividing vectors is possible?

There are some special cases where dividing vectors can be possible, such as when the vectors are parallel or when they are in the same direction. In these cases, dividing the magnitude of one vector by the magnitude of the other can give us a meaningful result.

5. Why do we need to know that dividing vectors is not possible?

Knowing that dividing vectors is not possible is important in understanding vector operations and in solving problems involving vectors. It helps us avoid making incorrect calculations and ensures that our solutions are mathematically accurate.

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