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I have the following question. In physics forces are vectors. Now I may write a force as \begin{equation} \mathbf{F} = F \mathbf{e}_F \end{equation} with $F$ denoting the length and $\mathbf{e}_F$ denoting the direction vector. But some forces are the result of a cross product (pseudo vectors). The length is then \begin{equation} \vert\mathbf{a}\vert \vert\mathbf{b}\vert \sin(\theta) \end{equation} with $\mathbf{a}, \mathbf{b}$ some vectors (maybe position and velocity) and $\theta$ the angle between them. However these vectors are also sometimes written in the first form. How can I check whether $F$ in the first form is the length of a cross product, or not?

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You can always choose $|a|, |b|, \theta$, that will give you that vector so... ...$F$ is always a length of infinite cross products. What are the restrictions? You must pick $a, b$ in a plane orthogonal to $F$!

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  • $\begingroup$ okay, but if I consider the length of the cross product, then a change of the sign is possible, but in the other formula a change would necessarily mean a change in the direction, right? $\endgroup$
    – Q.stion
    Jun 5, 2019 at 20:09
  • $\begingroup$ In the other formula, you can change the sign of $e_F$ by putting a minus before it or you can put a minus directly before $F$. This will change their direction. $\endgroup$ Jun 6, 2019 at 15:53
  • $\begingroup$ ...or you can swap $|b|$ with $|a|$. $\endgroup$ Jun 7, 2019 at 0:51

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