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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Vector Algebra
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If $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are non zero, non collinear vectors such that $\overrightarrow a\times\overrightarrow b=\overrightarrow b\times\overrightarrow c=\overrightarrow c\times\overrightarrow a$, then $\overrightarrow a+\overrightarrow b+\overrightarrow c=?$

$\begin{array}{1 1} 3\overrightarrow a \\ \overrightarrow 0 \\ unit \;vector \\ \overrightarrow a \end{array} $

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Given: $\overrightarrow a\times\overrightarrow b=\overrightarrow b\times\overrightarrow c$
$\Rightarrow\:\overrightarrow a\times\overrightarrow b=-\overrightarrow c\times\overrightarrow b$
$\Rightarrow\:\overrightarrow a\times\overrightarrow b+\overrightarrow c\times\overrightarrow b=0$
$\Rightarrow\:(\overrightarrow a+\overrightarrow c)\times\overrightarrow b=0$
$\Rightarrow\:\overrightarrow b $ is parallel to $\overrightarrow a+\overrightarrow c$
$\Rightarrow\:\overrightarrow a+\overrightarrow c=\lambda \overrightarrow b$
$\Rightarrow\:\overrightarrow c\times(\overrightarrow a+\overrightarrow c)=\lambda\overrightarrow c\times \overrightarrow b$
$\Rightarrow\:\overrightarrow c\times\overrightarrow a=\lambda\overrightarrow c\times \overrightarrow b$
$\Rightarrow\:\overrightarrow c\times\overrightarrow a=-\lambda\overrightarrow b\times \overrightarrow c$
But since it is given that $\overrightarrow c\times\overrightarrow a=\overrightarrow b\times \overrightarrow c$,
$\therefore\:\overrightarrow a+\overrightarrow c=- \overrightarrow b$
$\Rightarrow\:\overrightarrow a+\overrightarrow b+\overrightarrow c=\overrightarrow 0$
answered Nov 20, 2013 by rvidyagovindarajan_1

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