I know I can get the total angle of two vectors by the dot product of them both, but what if I just want to break down the angle into components of angle by X and angle by Y?
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It depends on what you mean by by X and by Y. If you mean get the angle on an axis of a vector, you will want to use inverse trigonometry, most likely
atan2.This reference is for C++, but the function should work the same way for most languages.