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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Vector Algebra
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If $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are unit vectors and $\overrightarrow a+\overrightarrow b+\overrightarrow c$ is also a unit vector and $\alpha,\beta,\gamma $ are angles between $\overrightarrow a\:and\:\overrightarrow b,\:\:\overrightarrow b\:and\:\overrightarrow c\:\:\overrightarrow c\:and\:\overrightarrow a$ respectively, then among the three angles,

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  • $\overrightarrow a.\overrightarrow b=|\overrightarrow a||\overrightarrow b|cos\theta$
Given: $|\overrightarrow a|=|\overrightarrow b|=|\overrightarrow c|=|\overrightarrow a+\overrightarrow b+\overrightarrow c|=1$
$|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+|\overrightarrow c|^2+2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)$
$\Rightarrow\:$ atleast one out of $cos\alpha,\:cos\beta,\:cos\gamma$ should be $ -ve$.
$\Rightarrow\:$ At least one out of the three angles should be obtuse.
answered Jan 6, 2014 by rvidyagovindarajan_1

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