Nabla, is defined as :
If you have a vector , then you can have only two multiplicative operations with this vector and the nabla vector. Those are the dot product ( )and the cross product ( ). And these represent divergence and curl respectively. It does not make sense to have scalar multiplcation between two vectors, which is what you are proposing with .
You may have scalar multiplication between two scalars, and scalar multiplcation between a scalar and a vector (a lá, , but you may not have scalar multiplication between two vectors.