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Linear Combination of Vectors

If a,b,c ar...

Question

If $a, b, c$ are unit vectors, then the maximum value of $| a + 2 b |^{2} + | b + 3 c |^{2} + | c + 4 a |^{2}$ is

A

50

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B

21

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C

48

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D

58

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Solution

The correct option is A $50$
Given,

${| a + 2 b |}^{2} + {| b + 3 c |}^{2} + {| c + 4 a |}^{2}$

$= {(a + 2 b)}^{2} + {(b + 3 c)}^{2} + {(4 a + c)}^{2}$

$= a^{2} + 4 a b + 4 b^{2} + b^{2} + 6 b c + 9 c^{2} + c^{2} + 8 a c + 16 a^{2}$

$= 17 a^{2} + 4 a b + 8 a c + 5 b^{2} + 10 c^{2} + 6 b c$

$= 50$

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