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$$W = ||V||(\cos(\varphi)\cdot \cos(\theta) - \sin(\varphi)\cdot\sin(\theta), \cos(\varphi)\cdot\sin(\theta) + \sin(\varphi)\cdot\cos(\theta))$$

$$= (v_1 \cos(\theta) - v_2 \sin(\theta), v_1 \sin(\theta) + v_2 \cos(\theta))$$

My question is what was the simplification step between the two lines? I can't seem to simplify the magnitude of the vector $V$ to get that final expression.

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I suppose $v_1$ and $v_2$ are the rectangular coordinates of vector $V$, and $\varphi$ is its angle in polar coordinates.

Thus $v_1=||V||\cos\varphi$ and $v_2=||V||\sin\varphi$.

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    $\begingroup$ Makes sense. Thanks. $\endgroup$ – dc3rd Nov 30 '14 at 23:27

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