Discussion Forum

If a, b and c are unit vectors  such that  a+b+c=0  and (a,b)=$\frac{\pi}{3}$, then

$|a \times b|+|b \times c|+|c \times a|=$

Answer:

Explanation:

 We have,

a,b,c are unit vector

$\therefore$   |a|=|b|=|c|=1  and a+b+c=0 , angle between a and b is $\frac{\pi}{3}$

Now,      a+b+c=0

$\Rightarrow$    $a \times (a+b+c)=0$

$\Rightarrow$  $a \times a+a \times b+a \times c=0$

$\Rightarrow$      $a \times b= c \times a$

$\Rightarrow$   $|a \times b|=|c \times a|$

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

$\therefore$  $|a \times b|+|b \times c|+|c \times a|$

                                      =$3 |a \times b|$

                                    =$3|a||b| \sin (a,b)$

                                      =$3 \times 1 \times 1 \times \sin \frac{\pi}{3}= \frac{3 \sqrt{3}}{2}$