If $\vec a, \vec b, \vec c$ are three vectors such that $\vec a + \vec b + \vec c = \vec 0$, then
Answer & explanation
Correct answer: option 2
We have,
$\vec a + \vec b + \vec c = \vec 0$ ...(i)
$⇒\vec a×(\vec a + \vec b + \vec c)=\vec a×\vec 0$ [Taking cross product with $\vec a$]
$⇒\vec a×\vec a+\vec a×\vec b+\vec a×\vec c=\vec 0$
$⇒\vec 0+\vec a×\vec b-\vec c×\vec a=0$
$⇒\vec a×\vec b=\vec c×\vec a$
Taking cross product of (i) with $\vec b$, we get $\vec a×\vec b=\vec b×\vec c$
$∴\vec a×\vec b =\vec b×\vec c = \vec c×\vec a$