Q: Assume such that , prove that the angle between and is equal to the angle between and

I've tried various things but I have yet to discover a good (formal) way of tackling this. I've written on the first line:

Now I've tried expanding this but I don't think it leads me anywhere so next I write the definition of projection vector:

Here it occurs to me that can be either positive or negative, therefore the projections will be either in the same direction or the opposite direction of and . So this leads me to believe that the projections do have the same angle by the congruence law of intersecting lines(?) Is this somewhat correct? Does it hold for , n>2? How do I format this into a formal proof?