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:

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