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 \vec U . Since you know \vec U is parallel to \vec Y, it's operating in the direction \pi/6. Break \vec U it into its x and y components:


\vec U= A \cos ( \frac {\pi} 6) \hat x + A \sin (\frac {\pi} 6) \hat y


The vector that is perpendicular to Y has magnitiude B, and is in the direction of \vec Y plus \pi/2 :


\vec V= B \cos ( \frac {2 \pi} 3) \hat x + B \sin (\frac {2 \pi} 3) \hat y


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