NIMCET 2009 Question Paper with solutions

Question : 22 of 120

Marks: +1, -0

\;\; a, b

and

c

are unit vectors, then

|a-b|^{2}+|b-c|^{2}+|c-a|^{2}

does not exceed

Solution:

|a+b+c|^{2}=|a^{2}+| b|+| c|^{2}+2(a ⋅ b+b ⋅ c+c ⋅ a)

⇒ 1+1+1+2(a ⋅ b+b ⋅ c+c ⋅ a) ≥ 0 \;\; (∵ ∣ a+b+c |^2 ≥ 0)

⇒ \;\; 2(a ⋅ b+b ⋅ c+c ⋅ a) ≥-3

⇒-2(a ⋅ b+b+c+c-a) ≤ 3

⇒ \;\; 6-2\(a ⋅ b+b ⋅ c+c ⋅ a) ≤9

⇒ 2(|a|^{2}+b|^{2}+|c|^{2}-a ⋅ b-b ⋅ c-c ⋅ a ≤ 9

(|a|^{2}+|b|^{2}-2 a ⋅ b)+(|b|^{2}+|c|^{2}-2b ⋅ c)

+(|c|^{2}+|a|-2 c.a) ≤ 9

⇒|a-b|^{2}+|b-c|^{2}+|c-a|^{2} ≤ 9

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