Two vectors with the same direction or in opposite direction we would say that they are parallel, independently that they have equal o different module.

If we take two parallel vectors and , and we locate them with the same origin, we see that we can pass from one to another multiplying by a scalar, that is to say,= k or = k '. Reciprocally, if = k or = k ', the two vectors have the same direction if k or k ' are positive, or opposite direction if k or k ' are negative; in any case, they are parallel.

Therefore, the parallelism condition of vectors is that they verifies = k or = k '. We observe that the scalars k and k ' are opposite one of the other.

If the components of and are =(a₁,a₂) and =(b₁,b₂), then the parallelism condition is (a₁,a₂)=k(b₁,b₂), that is to say:
                                                                     a₁= kb₁
                                                                     a₂= kb₂

isolating the k of two previous equalities and making equal the result, it is get the parallelism condition of two vectors given for its components:
                                                                   
that is to say, the components have to be proportional.