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+2ˆa1⋅ˆa2+2ˆa2⋅ˆa3+2ˆa3⋅ˆa1=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 = -32ˆa1⋅ˆa2+2ˆa2⋅ˆa3+2ˆa3⋅ˆa1=−3
Now
|hat a_1-hat a_2|^2+|hat a_2-hat a_3|^2+|hat a_3-hat a_1|^2|ˆa1−ˆa2|2+|ˆa2−ˆa3|2+|ˆa3−ˆa1|2
= (|hat a_1|^2+|hat a_2|^2-2hat a_1 cdot hat a_2)=(|ˆa1|2+|ˆa2|2−2ˆa1⋅ˆa2)
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