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Unit Vectors

If a, b and c...

Question

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 B 9
$| a - b |^{2} + | b - c |^{2} + | c - a |^{2} = 2 (a^{2} + b^{2} + c^{2}) - 2 (a . b + b . c + c . a) = 2 \times 3 - 2 (a . b + b . c + c . a) = 6 - (a + b + c)^{2} + a^{2} + b^{2} + c^{2} = 9 - | a + b + c |^{2} \leq 9$ .

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