Question

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

• A. $a\times b=b\times c=c\times a=0$
• B. $a\times b=b\times c=c\times a$ not equal to 0
• C. $a\times b=b\times c=a\times c$not equal to 0
• D. None of these

Answer 1

The correct option is B $a\times b=b\times c=c\times a$ not equal to 0

We have,
$a+b+c=0$ ...(A)
Taking cross product with a both sides of (A)
$a\times (a+b+c=0)$
$\Rightarrow a\times a+a\times b+a\times c=a\times 0$
$\Rightarrow 0+a\times b-c\times a=0$
$\Rightarrow a\times b=c\times a$ ...(i)
Taking cross product with b both sides of (A)
$b\times (a+b+c=0)$
$\Rightarrow b\times a+b\times b+b\times c=b\times 0$
$\Rightarrow -a\times b+0+b\times c=0$
$\Rightarrow a\times b=b\times c$ ...(ii)
From (i) and (ii)
$a\times b=b\times c=c\times a$ not equal to 0.
Option B is correct.