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I would like to show (I'm not yet sure if it's true, though), that any vector $v\in \mathbb{R}^3$ with $\|v\| = 1$ can be written as $\left(\cos(\beta)\sin(\alpha),\; \sin(\alpha)\sin(\beta), \; \cos^2\left(\frac{\alpha}{2}\right)-\sin^2\left(\frac{\alpha}{2}\right)\right)^T$.

Hereby, $0 \leq \alpha \leq \pi$, and $0 \leq \beta < 2 \pi$

Any ideas whether it's true or how to show it?

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    $\begingroup$ Don't you mean $\cos^2-\sin^2$ in the last one?. Anyways, take a look at this $\endgroup$
    – Pedro
    Jul 25, 2013 at 14:15
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    $\begingroup$ Change the first cos(alpha) to sin(alpha), and replace sin(alpha/2) by sin^2(alpha/2), and then it works. $\endgroup$ Jul 25, 2013 at 14:15
  • $\begingroup$ Yes, editing it. Thanks! Can you give me a hint on how it works? $\endgroup$ Jul 25, 2013 at 14:16

1 Answer 1

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Let $v=(v_1,v_2,v_3)$ have length 1.Let $\alpha$ be the angle between the vector $v$ and the vector $(0,0,1)$. Let $\beta$ be the angle between $(v_1,v_2,0)$ and $(1,0,0)$. We find that $v_1=sin(\alpha)cos(\beta),v_2=sin(\alpha)sin(\beta),v_3=cos(\alpha)=cos^2(\frac{\alpha}{2})-sin^2(\frac{\alpha}{2})$

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  • $\begingroup$ Please prefix trig.functions' names in $\LaTeX$ with a backslash, so that they render in upright font as functions' symbols: \sin → $\sin$ rather than as mutiplication of three variables, s, i & n: sin → $sin$. Same for $\cos$. $\endgroup$
    – CiaPan
    Jul 14, 2016 at 8:34

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