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}