Let $\vec a, \vec b, \vec c$ be unit vectors such that $\vec a + \vec b + \vec c = \vec 0$. Which one of the following is correct?

$\vec a×\vec b =\vec b×\vec c = \vec c×\vec a≠\vec 0$

We have,

$\vec a + \vec b + \vec c = \vec 0$

⇒ Unit vectors $\vec a, \vec b, \vec c$ represent three sides of a triangle taken in order

⇒ Triangle is equilateral.

We get,

$\vec a×\vec b =\vec b×\vec c = \vec c×\vec a$

Also, $\frac{1}{2}|\vec a×\vec b|=\frac{1}{2}|\vec b×\vec c|=\frac{1}{2}|\vec c×\vec a|$ = Area of triangle

$⇒\frac{1}{2}|\vec a×\vec b|=\frac{1}{2}|\vec b×\vec c|=\frac{1}{2}|\vec c×\vec a|=\frac{\sqrt{3}}{4}$

$∴\vec a×\vec b =\vec b×\vec c = \vec c×\vec a≠\vec 0$