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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Vector Algebra
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If $\overrightarrow a\:\:and\:\:\overrightarrow b$ are unit vectors and $\theta$ is angle between them then $\large\frac{1}{2}$$|\overrightarrow a-\overrightarrow b|$=?

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  • $|\overrightarrow a-\overrightarrow b|^2=(\overrightarrow a-\overrightarrow b).(\overrightarrow a-\overrightarrow b)$
  • $\overrightarrow a.\overrightarrow b=|\overrightarrow a||\overrightarrow b|cos\theta$
Given: $|\overrightarrow a|=|\overrightarrow b|=1$
$|\overrightarrow a-\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2-2\overrightarrow a.\overrightarrow b$
$=1+1-2cos\theta$
$\Rightarrow\:|\overrightarrow a-\overrightarrow b|^2=2-2cos\theta=2(1-cos\theta)$
$\Rightarrow\:|\overrightarrow a-\overrightarrow b|^2=4sin^2{\large\frac{\theta}{2}}$
or
$\Rightarrow\:|\overrightarrow a-\overrightarrow b|=2sin\large\frac{\theta}{2}$
i.e., $\large\frac{1}{2}$$|\overrightarrow a-\overrightarrow b|=sin\large\frac{\theta}{2}$
answered Nov 4, 2013 by rvidyagovindarajan_1
 

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