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

  1. 6
  2. 9
  3. 4
  4. 8

1 Answer
Ananda Dasgupta
Jun 22, 2018

9

Explanation:

#hat a_1+hat a_2+hat a_3=0 implies#

#(hat a_1+hat a_2+hat a_3)^2=0 implies#

#|hat a_1|^2+|hat a_2|^2+|hat a_3|^2+2hat a_1 cdot hat a_2+2hat a_2 cdot hat a_3+2hat a_3 cdot hat a_1=0#

Since the vectors are all of unit length, we have

#2hat a_1 cdot hat a_2+2hat a_2 cdot hat a_3+2hat 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#
#= (|hat a_1|^2+|hat a_2|^2-2hat a_1 cdot hat a_2)#
#qquad + (|hat a_2|^2+|hat a_3|^2-2hat a_2 cdot hat a_3)#
#qquad + (|hat a_3|^2+|hat a_1|^2-2hat a_3 cdot hat a_1)#
#= 6-(2hat a_1 cdot hat a_2+2hat a_2 cdot hat a_3+2hat a_3 cdot hat a_1)#
#=6-(-3) = 9#

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