IIT-JEE 2007 · Paper 2, Mathematics question 50

Cross with $\vec a$. From $\vec a+\vec b+\vec c=\vec0$: $\vec a\times\vec b+\vec a\times\vec c=\vec0$, so $\vec a\times\vec b=\vec c\times\vec a$. Crossing with $\vec b$ similarly gives $\vec a\times\vec b=\vec b\times\vec c$.

Nonzero. Three unit vectors adding to zero form an equilateral triangle, so any two are at 120°: $|\vec a\times\vec b|=\sin120°\ne0$.

So $\vec a\times\vec b=\vec b\times\vec c=\vec c\times\vec a\ne\vec0$. ((C) has $\vec a\times\vec c=-\vec c\times\vec a$, wrong sign; (D) fails since the three are equal.)

(B)