# Let $\overrightarrow{u},\overrightarrow{v}$ and $\overrightarrow{w}$ be vector such that $\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}=\overrightarrow{0}.$ If $|\overrightarrow{u}|=3, |\overrightarrow{v}|=4$ and $|\overrightarrow{w}|=5$ than find $\overrightarrow{u}.\overrightarrow{v}+\overrightarrow{v}.\overrightarrow{w}+\overrightarrow{w}.\overrightarrow{u}$

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• For any two vectors $\hat a \: and \: \hat b$ $(\hat a + \hat b)^2=(\hat a)^2+2\hat a.\hat b+(\hat b)^2=a^2+2\hat a.\hat b+b^2$ $(\hat a-\hat b)^2=a^2-2\hat a.\hat b+b^2$ $(\hat a+\hat b).(\hat a-\hat b)=a^2-b^2$

$\overrightarrow u + \overrightarrow v+\overrightarrow w=\overrightarrow 0$
$\therefore ( \overrightarrow u+\overrightarrow v+\overrightarrow w)^2= \overrightarrow 0.\overrightarrow 0=0$
$(\overrightarrow u+\overrightarrow v+\overrightarrow w).(\overrightarrow u+\overrightarrow v+\overrightarrow w)=0$
$\overrightarrow u^2+\overrightarrow v^2+\overrightarrow w^2+2\overrightarrow u.\overrightarrow v+2\overrightarrow v.\overrightarrow w+2\overrightarrow w.\overrightarrow u=0$
$9+16+25+2 [ \overrightarrow u.\overrightarrow v+\overrightarrow v.\overrightarrow w+\overrightarrow w.\overrightarrow u]=0$
$\therefore \overrightarrow u.\overrightarrow v+\overrightarrow v.\overrightarrow w+\overrightarrow w.\overrightarrow u= \large\frac{-50}{2} = -25$

answered Jun 2, 2013
edited Jun 3, 2013