Let \(\vec{r}\), \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are non-zero vectors such that \(\vec{r}\).\(\vec{a}\) = 0,

|\(\vec{r}\)x\(\vec{b}\)| = |\(\vec{r}\)||\(\vec{b}\)|, |\(\vec{r}\)x\(\vec{c}\)| = |\(\vec{r}\)||\(\vec{c}\)|, then [\(\vec{a}\) \(\vec{b}\) \(\vec{c}\)] is equal to : 

0

|\(\vec{r}\)x\(\vec{b}\)| = |\(\vec{r}\)||\(\vec{b}\)|

Thus, angle between \(\vec{r}\) and \(\vec{b}\) is \(\frac{\pi}{2}\)

\(\vec{r}\).\(\vec{a}\) = 0 ⇒ \(\vec{r}\) is ⊥ to \(\vec{a}\)

|\(\vec{r}\)x\(\vec{c}\)| = |\(\vec{r}\)|.|\(\vec{c}\)| ⇒ \(\vec{r}\) is ⊥ to \(\vec{c}\) 

Thus, \(\vec{r}\) is ⊥ to \(\vec{a}\), \(\vec{b}\), \(\vec{c}\).

Thus, \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are coplanar.

[\(\vec{a}\) \(\vec{b}\) \(\vec{c}\)] = 0