In a $3$-D vector space, does a vector $R$ being perpendicular to $A$, $B$, $C$ imply that $A$, $B$, $C$ are coplanar?

Question

In $3$-D vector space, If there's a vector $R$ which is given to be perpendicular to three others vectors $A , B , C$. Then can just this statement imply that vectors $A ,B ,C$ must be coplanar?

for Eg : Given info was $R\cdot A=0$ $$ |R\times B|=|R||B| $$ $$ |R\times C|=|R||C| $$ And then it was asked to find value of $[A B C ]$

Answer given was $0$.

Can anyone explain the reason behind this conclusion?

Answer 1

If $|R \times B| = |R||B|$ and $|R \times C| = |R||C|$ this would imply that $R || (B \times C)$ or $|R| = 0$

Since you know that $R.A = 0$, this would imply that $A.(B\times C) = 0$

The only case that would not be covered here is if $B,C$ are collinear. In that case $B\times C$ is $\vec{0}$, so it would still result in the value being 0