How do you divide a two or three component vector by another?
Or how do you take the same vector to the power of negative one?
I.E. :: V^(-1).
Thank you .
As far as I know, there is no such thing as vector division. You can divide by the length of a vector, which just ends up being scalar multiplication. What's your application? Why would you need to divide a vector by another?
[EDIT]: See here.
Note that Ackbeet's link refers to vectors that can be represented by complex numbers and quaternions which are specifically vectors of dimension 2 and 4. In general there is NO definition for multiplying two vectors together and so no definition for powers of vectors or vector division.
It may be that in a specific application, one can define a "type" of division. Or perhaps you are confusing division by the length of a vector with division by the vector.
Well, aside from the dot product and the cross product. The result of a dot product is a scalar, so right there you're stuck if you want to do anything more. Technically, you could do vector cross product after vector cross product. But a vector crossed with itself is the zero vector, so powers of a vector defined in terms of the cross product would get kinda uninteresting pretty quick.In general there is NO definition for multiplying two vectors together and so no definition for powers of vectors or vector division.
Cheers.
Yes, I completely forgot about the cross product. But that only applies to three dimensional vector spaces and, in any case, does not have an inverse. And I don't count the dot product- the result of a dot product is a scalar, not another vector, and so is not a "true" product.
assuming inner product as the definition of vector multipication, by the standard definiton of division, c divided by a is b st
a.b=c
However, as pointed out by AckBeet, the dot product is a scalar, and since there is no general definition of vector product such that ab=c, vector division is meaningless.
I forget, is there a general n dimensional definition of vector cross product? If so, then you could define divison as above. For three dimensions and the standard definition of cross product, c divided by a is b st aXb=c. But that doesn't work either since bXa = -aXb. However, you could define division by one or the other, and then you would have to investigate whether there is a solution fo every c and a, a unequal to zero.