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Unit Vectors

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Question

If $^a,^b$ and $^c$ are three unit vectors in the 3-D space, then the minimum value of $|^a +^b |^{2} + |^b +^c |^{2} + |^c +^a |^{2}$ is

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3 \sqrt{3}

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\frac{3}{2}

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Solution

The correct option is C 3
$|^a +^b |^{2} = 1 + 1 + 2 (^a .^b) |^b +^c |^{2} = 1 + 1 + 2 (^b .^c) |^c +^a |^{2} = 1 + 1 + 2 (^c .^a)$
and $(^a +^b +^c)^{2} = 1 + 1 + 1 + 2 (^a .^b +^b .^c +^c .^a) \geq 0$
$3 + 2 (^a .^b +^b .^c +^c .^a) \geq 0 \Rightarrow 6 + 2 (^a .^b +^b .^c +^c .^a) \geq 3 \Rightarrow |^a +^b |^{2} + |^b +^c |^{2} + |^c +^a |^{2} \geq 3$
$∴$ Minimum value =3

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If ^a,^b and ^c are three unit vectors in the 3-D space, then the minimum value of |^a+^b|2+|^b+^c|2+|^c+^a|2 is

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