Let $\vec a,\vec b,\vec c$ represent respectively $\vec{BC},\vec{CA}$ and $\vec{AB}$ where ABC is a triangle. Then,
Answer & explanation
Correct answer: option 3
We have,
$\vec{BC}+\vec{CA} =\vec{BA}$
$⇒\vec a+\vec b=-\vec c$
$⇒\vec a+\vec b+\vec c=\vec 0$
$⇒\vec a×(\vec a+\vec b+\vec c)=\vec a×\vec 0$
$⇒\vec a×\vec b+\vec a×\vec c=\vec 0⇒\vec a×\vec b=\vec c×\vec a$
Similarly, we have $\vec b×\vec c=\vec c×\vec a$
$∴\vec a× \vec b = \vec b× \vec c = \vec c× \vec a$
Clearly, $\vec a, \vec b,\vec c$ are coplanar vectors.
$∴[\vec a\, \vec b\, \vec c] = [\vec b\, \vec c\, \vec a]=[\vec c\, \vec a\, \vec b]=0$
So, option (4) is not correct.