If hata_1+hata_2+hata_3=0ˆa1+ˆa2+ˆa3=0 then the value of |hata_1-hata_2|^2+|hata_2-hata_3|^2+|hata_3-hata_1|^2|ˆa1ˆa2|2+|ˆa2ˆa3|2+|ˆa3ˆa1|2 is?

options are

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

1 Answer
Jun 22, 2018

9

Explanation:

hat a_1+hat a_2+hat a_3=0 impliesˆa1+ˆa2+ˆa3=0

(hat a_1+hat a_2+hat a_3)^2=0 implies(ˆa1+ˆa2+ˆa3)2=0

|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|ˆa1|2+|ˆa2|2+|ˆa3|2+2ˆa1ˆa2+2ˆa2ˆa3+2ˆa3ˆa1=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 = -32ˆa1ˆa2+2ˆa2ˆa3+2ˆa3ˆa1=3

Now

|hat a_1-hat a_2|^2+|hat a_2-hat a_3|^2+|hat a_3-hat a_1|^2|ˆa1ˆa2|2+|ˆa2ˆa3|2+|ˆa3ˆa1|2
= (|hat a_1|^2+|hat a_2|^2-2hat a_1 cdot hat a_2)=(|ˆa1|2+|ˆa2|22ˆa1ˆa2)
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