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?
Answer & explanation
Correct answer: option 2
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
$⇒\frac{1}{2}|\vec a×\vec b|=\frac{1}{2}|\vec b×\vec c|=\frac{1}{2}|\vec c×\vec a|=\frac{\sqrt{3}}{4}$
$∴\vec a×\vec b =\vec b×\vec c = \vec c×\vec a≠\vec 0$