# If hata_1+hata_2+hata_3=0 then the value of |hata_1-hata_2|^2+|hata_2-hata_3|^2+|hata_3-hata_1|^2 is?

## options are 6 9 4 8

Jun 22, 2018

9

#### Explanation:

${\hat{a}}_{1} + {\hat{a}}_{2} + {\hat{a}}_{3} = 0 \implies$

${\left({\hat{a}}_{1} + {\hat{a}}_{2} + {\hat{a}}_{3}\right)}^{2} = 0 \implies$

$| {\hat{a}}_{1} {|}^{2} + | {\hat{a}}_{2} {|}^{2} + | {\hat{a}}_{3} {|}^{2} + 2 {\hat{a}}_{1} \cdot {\hat{a}}_{2} + 2 {\hat{a}}_{2} \cdot {\hat{a}}_{3} + 2 {\hat{a}}_{3} \cdot {\hat{a}}_{1} = 0$

Since the vectors are all of unit length, we have

$2 {\hat{a}}_{1} \cdot {\hat{a}}_{2} + 2 {\hat{a}}_{2} \cdot {\hat{a}}_{3} + 2 {\hat{a}}_{3} \cdot {\hat{a}}_{1} = - 3$

Now

$| {\hat{a}}_{1} - {\hat{a}}_{2} {|}^{2} + | {\hat{a}}_{2} - {\hat{a}}_{3} {|}^{2} + | {\hat{a}}_{3} - {\hat{a}}_{1} {|}^{2}$
$= \left(| {\hat{a}}_{1} {|}^{2} + | {\hat{a}}_{2} {|}^{2} - 2 {\hat{a}}_{1} \cdot {\hat{a}}_{2}\right)$
$q \quad + \left(| {\hat{a}}_{2} {|}^{2} + | {\hat{a}}_{3} {|}^{2} - 2 {\hat{a}}_{2} \cdot {\hat{a}}_{3}\right)$
$q \quad + \left(| {\hat{a}}_{3} {|}^{2} + | {\hat{a}}_{1} {|}^{2} - 2 {\hat{a}}_{3} \cdot {\hat{a}}_{1}\right)$
$= 6 - \left(2 {\hat{a}}_{1} \cdot {\hat{a}}_{2} + 2 {\hat{a}}_{2} \cdot {\hat{a}}_{3} + 2 {\hat{a}}_{3} \cdot {\hat{a}}_{1}\right)$
$= 6 - \left(- 3\right) = 9$