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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Vector Algebra
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Q)

If $\overrightarrow a,\:\overrightarrow b,\:\overrightarrow c$ are unit vectors, then the maximum value of $|\overrightarrow a-\overrightarrow b|^2+|\overrightarrow b-\overrightarrow c|^2+|\overrightarrow c-\overrightarrow a|^2$ is ?

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A)
Given: $|\overrightarrow a|=|\overrightarrow b|=|\overrightarrow c|=1$
$|\overrightarrow a-\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2-2\overrightarrow a.\overrightarrow b$
$|\overrightarrow b-\overrightarrow c|^2=|\overrightarrow b|^2+|\overrightarrow c|^2-2\overrightarrow b.\overrightarrow c$
$|\overrightarrow c-\overrightarrow a|^2=|\overrightarrow c|^2+|\overrightarrow a|^2-2\overrightarrow c.\overrightarrow a$
$\therefore\:|\overrightarrow a-\overrightarrow b|^2+|\overrightarrow b-\overrightarrow c|^2+|\overrightarrow c-.\overrightarrow a|^2=6-2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)$
This will be maximum when $2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)$ is minimum.
We know that $|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2\geq 0$
$\Rightarrow\:|\overrightarrow a|^2+|\overrightarrow b|^2+|\overrightarrow c|^2+2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)\geq 0$
$\Rightarrow\:2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)\geq-3$
$i.e.,$ The least value of $ 2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)$ is $ -3$
$\therefore$ The maximum value of $|\overrightarrow a-\overrightarrow b|^2+|\overrightarrow b-\overrightarrow c|^2+|\overrightarrow c-.\overrightarrow a|^2$ is $ 6+3=9$
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