Given vectors and that and for some nonzero vector , prove that .

So I showed that if then , so now I assume are nonzero vectors. So and (where is the angle between and and is the angle between and ) and since all three vectors are nonzero, I can solve for one of the magnitudes and show that and then from there I get . Now, can I assume that the angles must be in the range of and therefore that . If that conclusion can be made, is it sufficient to show that if two vectors have the same magnitude and that the angles between them and the vector are the same, then they are the same vector? If not, how would I go about proving this? Thanks.