Clarify Right hand Rule

Question

I was just wondering whether in the right hand rule are all 3 vectors perpendicular to one another or is it simply one way, i.e. $A \times B=C$, would it be right to also say $C \times A=B$ and $C \times B=A$??

Answer 1

Given $a$ and $b$, $a\times b$ is perpendicular to both $a$ and $b$ regardless of whether $a$ and $b$ are perpendicular. The right-hand rule imposes an orientation on which direction is "up" and how the vectors are oriented relative to one another. See this and this.

So, from your example, if we know $c$ and $a$, then we know what direction $c\times a$ points relative to $c$ and $a$, by following the right hand rule.

Similarly, given $c$ and $b$, we know what direction $c\times b$ points relative to $c$ and $b$.