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

If $\overrightarrow a,\:\overrightarrow b,\:\overrightarrow c$ are any three vectors such that $\overrightarrow a\times\overrightarrow b=\overrightarrow c\:\:and\:\:\overrightarrow b\times\overrightarrow c=\overrightarrow a$, then the magnitude of $\overrightarrow b$ is?

$\begin{array}{1 1} (A) |\overrightarrow b|=|\overrightarrow a| \\ (B) |\overrightarrow b|=|\overrightarrow c| \\ (C) |\overrightarrow b|=|\overrightarrow a|=|\overrightarrow c| \\ (D) |\overrightarrow b|=1 \end{array} $

1 Answer

Given: $\overrightarrow a\times\overrightarrow b=\overrightarrow c\:\:and\:\:\overrightarrow b\times\overrightarrow c=\overrightarrow a$
$\Rightarrow\:\overrightarrow c$ is $\perp$ to $\overrightarrow a\:\:and\:\:\overrightarrow b$ and
$\overrightarrow a$ is $\perp$ to $\overrightarrow b\:\:and\:\:\overrightarrow c$
$\Rightarrow\:\overrightarrow a\perp \overrightarrow b\perp \overrightarrow c$
$\Rightarrow\:|\overrightarrow c|=|\overrightarrow a||\overrightarrow b|sin\large\frac{\pi}{2}$$=|\overrightarrow a||\overrightarrow b| \:\:and$
$|\overrightarrow a|=|\overrightarrow b||\overrightarrow c|$
$\Rightarrow\:|\overrightarrow a|=|\overrightarrow b|.|\overrightarrow a|.|\overrightarrow b|$
$\Rightarrow\:1=|\overrightarrow b|^2$
or $|\overrightarrow b|=1$
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