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

If $|\overrightarrow a|=|\overrightarrow c|=1, \:|\overrightarrow b|=4,\:\overrightarrow a\times\overrightarrow b=2\overrightarrow a\times\overrightarrow c\:and\:$ if the angle between $\overrightarrow b\:and\:\:\overrightarrow c$ is $cos^{-1}\large\frac{1}{4}$ then $\overrightarrow b-2\overrightarrow c=?$

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A)
Given: $\overrightarrow a\times\overrightarrow b=2\overrightarrow a\times\overrightarrow c$
$\Rightarrow\:\overrightarrow a\times(\overrightarrow b-2\overrightarrow c)=\overrightarrow 0$
$\Rightarrow\:\overrightarrow b-2\overrightarrow c=\lambda\overrightarrow a$
$\Rightarrow\:|\overrightarrow b-2\overrightarrow c|^2=|\lambda|^2 |\overrightarrow a|^2$
$\Rightarrow\:|\overrightarrow b|^2+4|\overrightarrow c|^2-4\overrightarrow b.\overrightarrow c=|\lambda|^2|\overrightarrow a|^2$
$\Rightarrow\:16+4-4.4.1.\large\frac{1}{4}=|\lambda|^2$
(Since it is given that $cos\theta=\large\frac{1}{4}$$,\:|\overrightarrow a|=|\overrightarrow c|=1\:and\:|\overrightarrow b|=4$)
$\Rightarrow\:\lambda^2=16\:\:or\:\:\lambda=\pm4$
$\therefore\:\overrightarrow b-2\overrightarrow c=\pm4\overrightarrow a$
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