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!