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If $\overrightarrow{a}+{b}+{c}=0,|\overrightarrow{a}|=3, |\overrightarrow{b}|=4,|\overrightarrow{c}|=5$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

\[\begin{array} {1 1}(1)\frac{\pi}{6} &(2)\frac{2\pi}{3}\\(3)\frac{5\pi}{3}&(4)\frac{\pi}{2}\end{array}\]

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Given $\overrightarrow{a}+{b}+{c}=0,|\overrightarrow{a}|=3, |\overrightarrow{b}|=4,|\overrightarrow{c}|=5$
$\overrightarrow{a}+\overrightarrow{b}=-\overrightarrow{c}$
$\overrightarrow{a}+\overrightarrow{b}=(-\overrightarrow{c})^2$
$\overrightarrow{a^2}+\overrightarrow{b^2}+2.\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{c^2}$
$\overrightarrow{|a|^2}+\overrightarrow{|b|^2}+2.\overrightarrow{|a|}.\overrightarrow{|b|} \cos \theta=\overrightarrow{|c|^2}$
$3^2+4^2+2 \times 3 \times 4 \cos \theta=5^2$
$9+16+24 \cos \theta=25$
$24 \cos \theta=0$
$\cos \theta=0$
$\theta =\large\frac{\pi}{2}$
Hence 4 is the correct answer.
answered May 5, 2014 by meena.p
 

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