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Unit Vectors

If a⃗, b⃗ and...

Question

If $\to a$ , $\to b$ and $\to c$ are unit vectors, then the maximum value of $| a - b |^{2} + | b - c |^{2} + | c - a |^{2}$ is

A

1

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B

4

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C

9

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Solution

The correct option is C 9
$| a - b |^{2} + | b - c |^{2} + | c - a |^{2} = 2 (a^{2} + b^{2} + c^{2}) - 2 (a . b + b . c + c . a) = 2 \times 3 - 2 (a . b + b . c + c . a) = 6 - (a + b + c)^{2} + a^{2} + b^{2} + c^{2} = 9 - | a + b + c |^{2}$

The minimum value of $| a + b + c |$ is 0. Hence, the maximum value of $| a - b |^{2} + | b - c |^{2} + | c - a |^{2}$ is 9 - 0 which is 9.

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