If →a,→b,→c are unit vectors, then ∣∣∣→a−→b∣∣∣2+∣∣∣→b−→c∣∣∣2+∣∣→c−→a∣∣2 does not exceed
A
4
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B
9
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C
8
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D
6
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Solution
The correct option is D9 ∣∣∣→a+→b+→c∣∣∣≥0 ∣∣∣→a+→b+→c∣∣∣2=∣∣→a∣∣2+∣∣∣→b∣∣∣2+∣∣→c∣∣2+2(→a.→b+→b.→c+→c+→a)≥0 ⇒∣∣→a∣∣2+∣∣∣→b∣∣∣2+∣∣→c∣∣2+2(→a.→b+→b.→c+→c+→a)≥0 Since →a,→b,→c are unit vectors we have 3+2(→a.→b+→b.→c+→c+→a)≥0 ∴→a.→b+→b.→c+→c.→a≥−32 .............(1) ∣∣∣→a−→b∣∣∣2+∣∣∣→b−→c∣∣∣2+∣∣→c−→a∣∣2 =2∣∣→a∣∣2+2∣∣∣→b∣∣∣2+2∣∣→c∣∣2−2(→a.→b+→b.→c+→c.→a) =2+2+2−2(→a.→b+→b.→c+→c.→a) =6−2(→a.→b+→b.→c+→c.→a) =6−2(−32)=6+3=9 using (1)