So I have these two vectors
X=3 {Cos[\[Pi]/3],Sin[\[Pi]/3]};
Y={Cos[\[Pi]/6],Sin[\[Pi]/6]};
and I want to obtain two other vectors U and V such that obey the following conditions:

U parallel to Y
V perpendicular to Y
X = U+V

I have tried many ways but still can't get the answer for U and V, please help me on this!

Let A equal the magnitude of . Since you know is parallel to , it's operating in the direction . Break it into its x and y components:



The vector that is perpendicular to Y has magnitiude B, and is in the direction of plus :



Now add the x components of the and vectors together, and set this equal to the x component of your vector, and do the same for the y components. This will give you two equations in 2 unknowns; solve for A and B.