<s>[INST] Three unit vectors whose sum is zero Let $\overrightarrow a ,\overrightarrow b ,\overrightarrow c $ be unit vector such that $\overrightarrow a  + \overrightarrow b  + \overrightarrow c  = 0$. Which of the following is correct ?
(A) $\overrightarrow a  \times \overrightarrow b  = \overrightarrow b  \times \overrightarrow c  = \overrightarrow c  \times \overrightarrow a  = \overrightarrow 0 $
(B) $\overrightarrow a  \times \overrightarrow b  = \overrightarrow b  \times \overrightarrow c  = \overrightarrow c  \times \overrightarrow a  \ne \overrightarrow 0 $
(C) $\overrightarrow a  \times \overrightarrow b  = \overrightarrow b  \times \overrightarrow c  = \overrightarrow a  \times \overrightarrow c  \ne \overrightarrow 0 $
(D) $\overrightarrow a  \times \overrightarrow b ,\overrightarrow b  \times \overrightarrow c ,\overrightarrow c  \times \overrightarrow a $ are mutually perpendicular
The official answer is (B).
My approach is s follow
$\overrightarrow a  + \overrightarrow b  + \overrightarrow c  = 0$
$\overrightarrow a  \times \overrightarrow b  + \overrightarrow b  \times \overrightarrow b  + \overrightarrow c  \times \overrightarrow b  = 0$
$\overrightarrow a  \times \overrightarrow b  =  - \overrightarrow c  \times \overrightarrow b  = \overrightarrow b  \times \overrightarrow c $
$\overrightarrow a  \times \overrightarrow c  + \overrightarrow b  \times \overrightarrow c  + \overrightarrow c  \times \overrightarrow c  = 0 \Rightarrow \overrightarrow b  \times \overrightarrow c  = \overrightarrow c  \times \overrightarrow a $.
Now I am confused between option (A) and (B) if $\overrightarrow a ,\overrightarrow b ,\overrightarrow c $ are parallel to each other then option (A) is correct or option (B)
Vector being a unit vector plays any vital role in deciding between (A) and (B)
 [/INST] (A) would imply the vectors are pairwise parallel or antiparallel, so their sum is of length $1$ or $3$.
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