why is vector division undefined?

Division basically refers to multiplication of reciprocal. For example a/b is same as a*1/b or we can say, is same as a*b^-1, which is "a" multiplied to the inverse of "b". There is no way of finding an inverse to a vector, and even if you could, you cant necessarily know if division in case of vectors implies left multiplication of inverse or right multiplication of inverse as we know that multiplication is not commutative in case of vectors i.e. AB≠BA where A,B are vectors. Hence, division of vectors is not defined.

If A and B are vectors, A/B could be AB^-1 or it could be B^-1A. Which way we mean isnt certain, and commutativity is an issue. And then finding b^-1 is also an issue. Therefore A/B cannot be found out.

as a vector has direction

when direction can be multiplied.then it should also be divided.