<s>[INST] Least value of unit vector $|a+b|^2+|b+c|^2+|c+a|^2$ If $a, b, c$ are unit vectors, then least value of $|a+b|^2+|b+c|^2+|c+a|^2$ will be equal to 
(1) 1
(2) 3
(3) 9
(4) 12  
If am using the concept $a=c=-b$, I am getting the answer 4, but not matching with options provided
 [/INST] Let $a,b,c$ be three unit vectors then $$(a+b+c)^2\ge 0 \implies a^2+b^2+c^2+2(a,b+b.c+c.a) \ge 0$$ $$ \implies 2(a.b+b.c+c.a) \ge -3 ~~~(1)$$
Then $$|a+b|^2+|b+c|^2+|c+a|^2=2(a^2+b^2+c^2)+2(a.b+b.c+c.a) \ge 6-3=3.$$
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