If →a→b, and →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 B9 ∣∣∣→a−→b∣∣∣2=∣∣→a∣∣2+∣∣∣→b∣∣∣2−2→a.→b∣∣∣→b−→c∣∣∣2=∣∣∣→b∣∣∣2+∣∣→c∣∣2−2→b.→c∣∣→c−→a∣∣2=∣∣→c∣∣2+∣∣→a∣∣2−2→c.→a∣∣∣→a−→b∣∣∣2+∣∣∣→b−→c∣∣∣2+∣∣→c−→a∣∣2=2(∣∣→a∣∣2+∣∣∣→b∣∣∣2+∣∣→c∣∣2−→a.→b−→b.→c−→c.→a)=2(3−→a.→b−→b.→c−→c.→a)=6−2(→a.→b+→b.→c+→c.→a)(→a+→b+→c)≥0∣∣→a∣∣2+∣∣∣→b∣∣∣2+∣∣→c∣∣2+2(→a.→b+→b.→c+→c.→a)≥0→a.→b+→b.→c+→c.→a≥−32⇒−2(→a.→b+→b.→c+→c.→a)≤3∣∣∣→a−→b∣∣∣2+∣∣∣→b−→c∣∣∣2+∣∣→c−→a∣∣2≤6+3≤9OptionBiscorrect.