There are no nasty surprises here. All that happens when I multiply by a scalar is that the magnitude of the vector is multiplied by this number. In general the physical dimension associated with the magnitude of the vector can change. If the scalar is positive then the direction of the vector is unchanged. If it is negative then the new vector points in the opposite direction. Of course I can also define division by a scalar as multiplication by the inverse of that scalar. An important type of vector called a unit vector come from this. If I divide a vector by its magnitude then I am left with a vector of magnitude unity. Such a vector is called a unit vector and it simply points out a direction.