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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Vector Algebra
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If $ |\overrightarrow a+\overrightarrow b|=|\overrightarrow a|$, then the angle between $2\overrightarrow a+\overrightarrow b$ and $\overrightarrow b$ is ?

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Given: $|\overrightarrow a+\overrightarrow b|=|\overrightarrow a|$
$\Rightarrow\:(\overrightarrow a+\overrightarrow b).(\overrightarrow a+\overrightarrow b)=\overrightarrow a.\overrightarrow a$
$\Rightarrow\:\overrightarrow a.\overrightarrow a+2\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow b=\overrightarrow a.\overrightarrow a$
$\Rightarrow\:2\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow b=0$
Angle between $2\overrightarrow a+\overrightarrow b$ and $\overrightarrow b$ is given by
$cos\theta=\large\frac{(2\overrightarrow a+\overrightarrow b).\overrightarrow b}{|2\overrightarrow a+\overrightarrow b||\overrightarrow b|}$
$=\large\frac{2\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow b}{|2\overrightarrow a+\overrightarrow b||\overrightarrow b|}$$=0$
$\therefore$ the angle between $2\overrightarrow a+\overrightarrow b$ and $\overrightarrow b$ is $\large\frac{\pi}{2}$
answered Nov 30, 2013 by rvidyagovindarajan_1

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