I am trying to figure out the effect of multiplying a complex vector in C^2 space with a complex scalar. The theory says this produces a parallel vector like the one in real vector space case. For example. If I multiply a real vector [a,b] by a real scalar c, it will be [ac,bc], that is a vector parallel to [a,b]. Similarly, if I multiply a complex vector [a+ib,c+id] by a complex scalar (lets say e+if), will the new vector be parallel to [a+ib,c+id]? How? Can you describe this geometrically?