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Home  >>  CBSE XII  >>  Math  >>  Vector Algebra
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Let $ \bar{a},\bar{b},\;and\;\bar{c}$ be three vectors such that $|\bar{a}|=3,|\bar{b}|=4,|\bar{c}|=5$ and each one of them being perpendicular to sum of the other two, find $ |\bar{a}+\bar{b}+\bar{c}|$

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Toolbox:
  • If two vectors $\overrightarrow a\:and\:\overrightarrow b $ are perpendicular then $\overrightarrow a.\overrightarrow b=0$.
  • $|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+|\overrightarrow c|^2$ $+2\overrightarrow a.\overrightarrow b+2\overrightarrow b.\overrightarrow c+2\overrightarrow c.\overrightarrow a$
Given $|\overrightarrow a|=3,|\overrightarrow b|=4\:and\:|\overrightarrow c=5$ and
$\overrightarrow a\:is\:\perp\:to\:\overrightarrow b+\overrightarrow c$, $\overrightarrow b\:is\:\perp\:to\:\overrightarrow c+\overrightarrow a\:and$
$\overrightarrow c\:is\:\perp\:to\:\overrightarrow a+\overrightarrow b$
$\Rightarrow\:\overrightarrow a.(\overrightarrow b+\overrightarrow c)=0,$
$\overrightarrow b.(\overrightarrow c+\overrightarrow a)=0$ and
$\overrightarrow c.(\overrightarrow a+\overrightarrow b)=0$
Adding three of the above equations we get
$\Rightarrow\:\overrightarrow a.(\overrightarrow b+\overrightarrow c)+\overrightarrow b.(\overrightarrow c+\overrightarrow a)+\overrightarrow c.(\overrightarrow a+\overrightarrow b)=0,$
$\Rightarrow\:2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)=0$
We know that
$|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+|\overrightarrow c|^2$+$2\overrightarrow a.\overrightarrow b+2\overrightarrow b.\overrightarrow c+2\overrightarrow c.\overrightarrow a$
$\Rightarrow\:|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+|\overrightarrow c|^2+0$
Because $2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)=0$
Substituting the values of $|\overrightarrow a|,|\overrightarrow b|\:and\:|\overrightarrow c|$,we get
$\Rightarrow\:|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2=3^2+4^2+5^2=50$
$\Rightarrow\:|\overrightarrow a+\overrightarrow b+\overrightarrow c|=\sqrt{50}=5\sqrt2$

 

answered Apr 20, 2013 by rvidyagovindarajan_1
 

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