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

If $\overrightarrow a=\hat i+\hat j,\:\:\overrightarrow b=2\hat j-\hat k$ and $\overrightarrow r\times \overrightarrow a=\overrightarrow b\times\overrightarrow a,\:\:\overrightarrow r\times\overrightarrow b=\overrightarrow a\times\overrightarrow b$ then a unit vector in the direction of $\overrightarrow r$ is ?

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A)
Given: $\overrightarrow r \times\overrightarrow b=\overrightarrow a\times\overrightarrow b\:\:and\:\:\overrightarrow r \times\overrightarrow a=\overrightarrow b\times\overrightarrow a$
$\Rightarrow\:\overrightarrow r\times\overrightarrow b=-\overrightarrow r\times\overrightarrow a$
$\Rightarrow\:\overrightarrow r\times (\overrightarrow a+\overrightarrow b)=0$
$\Rightarrow\:\overrightarrow r\:\:is\:\:$ parallel to $\overrightarrow a+\overrightarrow b$
$\Rightarrow\:\overrightarrow r=\lambda (\overrightarrow a+\overrightarrow b)$
$i.e.,\:\:\overrightarrow r=\lambda(\hat i+3\hat j-\hat k)$
$\Rightarrow\:\hat r=\large\frac{1}{\sqrt {11}}$$(\hat i+3\hat j-\hat k)$
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