How to figure out x and y components of a vector

Question

Given a vector $\vec{v}$ and an angle $\alpha$ between the vector and $Oy$ or $Ox$ what is the quickest way to figure out the projections, or the $x$ and $y$ components?

Answer 1

If the angle $\alpha$ is oriented counterclockwise (assuming right handed coordinates) from $Ox$ to $\vec{v}$, then the $x$ and $y$ coordinates are $|v|\cos(\alpha)$ and $|v|\sin(\alpha)$ resp, with proper signs. Similar results hold if the angle is from $Oy$, or orientations are different.