Question

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

• A. $a\times b=b\times c=c\times a=0$
• B. $a\times b=b\times c=c\times a\ne 0$
• C. $a\times b=b\times c=a\times c\ne 0$
• D. $a\times b,b\times c,c\times a$ are mutually perpendicular

Answer 1

The correct option is B $a\times b=b\times c=c\times a\ne 0$

$a+b+c=0\Rightarrow a,b,c$ are coplanar. Also no two of $a,b,c$ can be parallel, for if a is parallel to $b,$
then $a=kb$ but $a,b$ are unit vectors so $k=\pm 1.$ If $k=1$ we get $a=b$ and so $2a+c=0$
$\Rightarrow \left|c\right|=2\left|a\right|=2,$ which is contradiction the assumption that $c$ is a unit vector.
If $k=-1,$ we get $a+b=0\Rightarrow c=0$
which contradicts the assumption that $c$ is a unit vector. Therefore $a$ is not parallel to $b$
$\Rightarrow a\times b\ne 0$. Also
$a+b+c=0$

$\Rightarrow a\times (a+b+c)=0$
$\Rightarrow a\times b=c\times a$
Similarly, $a\times b=b\times c$
Thus $a\times b=b\times c=c\times a\ne 0$