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