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I have an equation: $\vec{v}=\vec{w}\times \vec{r}$ How to I separate the $\vec{w}$ and write it in terms of $\vec{v}$ and $\vec{r}$?

I tried to re-arrange this equation so that I could find the direction of $\vec w$ as the direction of other two are given.

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Rifat
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  • I tried to re-arrange this equation so that I could find the direction of $\vec{w}$, as the direction of other two are given – Rifat Jun 27 '20 at 06:54
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    In general it's not possible because the map $\vec{w} \mapsto \vec{w} \times \vec{r}$ is not injective. (i.e it is possible to have two different $\vec{w}$ which after you take cross product, give the same result). For example, we have that for any $t \in \Bbb{R}$, $(e_x + t e_y) \times e_y = e_z$, so it possible to have infinitely many $w$ with the same output. More generally, for any $t \in \Bbb{R}$, we have $(\vec{w} + t \vec{r}) \times \vec{r} = \vec{w} \times \vec{r} = v$. i.e we still have infinitely many things with the same output. – peek-a-boo Jun 27 '20 at 06:54
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    $\vec w$ is not uniquely defined in terms of $\vec v$ and $\vec r$. – Andrew Chin Jun 27 '20 at 06:54
  • Sometimes a search is all it takes. – Andrew Chin Jun 27 '20 at 06:55
  • @AndrewChin does it have two or multiple possibilities? – Rifat Jun 27 '20 at 06:56
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    @AndrewChin here's the origonal problem https://ibb.co/tMGs97Y – Rifat Jun 27 '20 at 06:59
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    @Rifat You are complicating the problem too much. Both angular velocity and angular acceleration are along $z$ axis (can be either positive or negative) – Andrei Jun 27 '20 at 07:26
  • @Andrei thank you ♥ – Rifat Jun 27 '20 at 07:42

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