If $\vec{a}$ x $\vec{b}$ = $\vec{c}$ and $\vec{b}$ x $\vec{c}$ = $\vec{a}$, then : 

$\vec{a}$, $\vec{b}$, $\vec{c}$ are orthogonal in pairs and |$\vec{a}$| = |$\vec{c}$| = |$\vec{b}$| = 1

$\vec{a}$ x $\vec{b}$ = $\vec{c}$ is ⊥ to both $\vec{a}$ and $\vec{b}$

$\vec{b}$ x $\vec{c}$ = $\vec{a}$ is ⊥ to both $\vec{b}$ and $\vec{c}$

Thus, $\vec{a}$, $\vec{b}$, $\vec{c}$ form an orthogonal system.

Taking mode of both sides of given relation |$\vec{a}$||$\vec{b}$| Sin$\frac{\pi}{2}$ = |$\vec{c}$| and |$\vec{b}$||$\vec{c}$| Sin $\frac{\pi}{2}$ = |$\vec{a}$|

Putting for |$\vec{c}$|, we get |$\vec{b}$|=1

⇒ |$\vec{a}$| = |$\vec{c}$| = |$\vec{b}$| = 1