Expressing a variable from cross product

[v-v]

Can I use cyclic rotation in $$\vec{a}$$ = $$\vec{b}$$ x $$\vec{c}$$ and say:
$$\vec{c}$$ = $$\vec{a}$$ x $$\vec{b}$$
$$\vec{b}$$ = $$\vec{c}$$ x $$\vec{a}$$

for any vectors $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ or only if they are perpendicular to each other?

If it's only a special case: is there a way to express $$\vec{b}$$ and $$\vec{c}$$ from the previous equation?

(I'm asking because of the equation in electromagnetism that says $$\vec{E}$$=c($$\vec{B}$$x$$\vec{U}$$) where I might need to find any of the 3 vectors from the other two)

I hope writing down the matrices, finding the inverses and solving a matrix equation is not the only way.

Thank you :)

 

[HallsofIvy]

Remember that the cross product of any two vectors is perpendicular to the two vectors. If $\vec{a}= \vec{b}\times\vec{c}$ with $\vec{b}$ and $\vec{c}$ not perpendicular, then we cannot have \vec{b}= \vec{c}\times\vec{a}[/itex] because that says $\vec{b}$ is perpendicular to $\vec{c}$ nor can $\vec{c}= \vec{a}\times\vec{b}$ because then $\vec{c}$ would be perpendicular to $\vec{b}$.

 

[v-v]

Thank you :)

 

[Rasalhague]

You can explore the various possibilities with this handy identity:

$$\left ( \textbf{a} \times \textbf{b} \right ) \times \textbf{c} = \left ( \textbf{a} \cdot \textbf{c} \right ) \textbf{b} - \left ( \textbf{b} \cdot \textbf{c} \right ) \textbf{a}$$

 

[CellCoree]

I would like to clarify that the cross product is a mathematical operation between two vectors that results in a vector perpendicular to both of the original vectors. In the equation \vec{a} = \vec{b} x \vec{c}, the vector \vec{a} is the cross product of \vec{b} and \vec{c}. This means that \vec{a} is perpendicular to both \vec{b} and \vec{c}.

In general, the cross product is not commutative, meaning that the order of the vectors matters. So, if we have \vec{a} = \vec{b} x \vec{c}, it does not necessarily mean that we can also write \vec{c} = \vec{a} x \vec{b} or \vec{b} = \vec{c} x \vec{a}. These equations are only valid if \vec{a}, \vec{b}, and \vec{c} are all perpendicular to each other.

In the context of electromagnetism, the equation \vec{E}=c(\vec{B}x\vec{U}) is a special case where \vec{E}, \vec{B}, and \vec{U} are all perpendicular to each other. This means that we can use the cross product to find any of the three vectors from the other two.

In general, if you want to express \vec{b} or \vec{c} from the equation \vec{a} = \vec{b} x \vec{c}, you can use the properties of the cross product to write \vec{b} = \frac{\vec{a} x \vec{c}}{|\vec{c}|} and \vec{c} = \frac{\vec{a} x \vec{b}}{|\vec{b}|}. However, this will only be valid if \vec{a}, \vec{b}, and \vec{c} are all perpendicular to each other.

In conclusion, the use of the cross product in expressing variables is limited to special cases where the vectors involved are perpendicular to each other. In other cases, we cannot simply use cyclic rotation to express one vector from the other two. Matrix equations and finding inverses may be necessary in those cases.