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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?

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You should use Pythagoras theorem. – Pragabhava Oct 3 '12 at 5:41
up vote 1 down vote accepted

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.

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