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Unit Vectors

If a,b and ...

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

Hence, option B.

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