Question

If $a$, $b$ and $c$ are unit vectors such that $a+b+c=0$. Then, which of the following is correct?

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

Answer 1

The correct option is B

$a\times b=b\times c=c\times a\ne 0$

Given, $a+b+c=0$

Thus, $a,b,c$ are coplanar.

And also no two vectors among $a,b,c$ are parallel, otherwise it won’t satisfy the given condition.

If we say, $a$ is parallel to $b$, then

$a=kb$ [since $a$ and $b$ are unit vectors so $k=\pm 1$.]

Now, if $k=1$, then

$a=b$

Thus, $2a+c=0$, then [from given equation]

$\left|c\right|,\left|a\right|=2$, which is not possible, since $c$ is a unit vector.

So $a$ is not parallel to $b$ .

$a\times b\ne 0$

Also, $a+b+c=0$

$a\times \left(a+b+c\right)=0$

$a\times b=a\times c$

Similarly, $a\times b=b\times c$

Therefore, $a\times b=b\times c=c\times a\ne 0$

Hence, the correct option is $\left(B\right)$.