What is vector division?

Question

My question is: We have addition, subtraction and muliplication of vectors. Why cannot we define vector division? What is division of vectors?

Answer 1

I assume that by "multiplication of vectors" you mean the dot product and the cross product. So if vector division did exist, it would have to have to be the inverse of at least one of those.

First let's take a look at dot product. If V, M, and K are vectors, then saying that V/M = K should imply that K*M = V. However, with dot product, K*M would be a scalar, just a normal lonely number, and not a vector, so that doesn't work.

So let's take a look at cross products. If vector division existed, I think it would be fair to expect that for a vector V, V/V = 1. This wouldn't be the number 1, but a vector 1, so that 1×<a,b> = <a,b>, since you can only define the cross product between two vectors (and taking the cross product of two vectors always gives a vector as an answer). However, there is identity element with cross products -- there is not vector to act like 1, no vector that you can take the cross product with any vector V and always get V as the answer. This also means you wouldn't be able to define V/K as V×1/K, since 1 doesn't exist with cross products.

I hope this answers your question. As a final note, you could define division with just scalar multiplication, as in, you could have <4,10>/2, since that would just be <4,10>*(1/2)= <2,5>, but you probably already knew that. Oh, and as a final, final note, though we don't have division with vectors, we do have division sometimes with matrices, as long as the matrices in question have inverses. Ok, that is all for me. Have a great day!

Answer 2

Dividing one quantity, say x, by another, say y, asks the question, "By what do we need to multiply y to get x?" In the case of numbers (scalars), the answer is also a scalar, as long as y is not zero. In order for the answer to be a scalar in the case of vectors, each component of x would have to be the same multiple of the corresponding component of y. If this is not the case, then we would need to multiply y by a matrix in order to obtain x: Ay=x. That means that the dot product of the first row of A times the column vector y must equal the first component of x, the dot product of the second row of A times y must equal the second component of x, and so on. There are many ways that each component of x can be formed as the dot product of a row vector times y, so the answer would not be unique. That is why the division of vectors is not uniquely defined. Instead, we must describe exactly what we mean, such as by supplying a specific matrix A and saying that Ay=x, in which case we can use the inverse matrix (if it exists) to express y as A^-1x.