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If $|\overrightarrow a+\overrightarrow b|=\sqrt {29}$ and $\overrightarrow a\times (2\hat i+3\hat j+4\hat k)=(2\hat i+3\hat j+4\hat k)\times \overrightarrow b,$ then $(\overrightarrow a+\overrightarrow b).(-7\hat i+2\hat j+3\hat k)=?$

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  • If $\overrightarrow a\times \overrightarrow b =0$, then $\overrightarrow a=\lambda \overrightarrow b$
Given : $\overrightarrow a\times (2\hat i+3\hat j+4\hat k)= (2\hat i+3\hat j+4\hat k)\times\overrightarrow b$
$\Rightarrow\:(\overrightarrow a+\overrightarrow b)\times (2\hat i+3\hat j+4\hat k)=0$
$\Rightarrow\: \overrightarrow a+\overrightarrow b=\lambda (2\hat i+3\hat j+4\hat k)$
Since it is given that $| \overrightarrow a+\overrightarrow b|=\sqrt {29}$,
$\lambda\sqrt {4+9+16}=\sqrt {29}$
$\Rightarrow\:(\overrightarrow a+\overrightarrow b).(-7\hat i+2\hat j+3\hat k)=(2\hat i+3\hat j+4\hat k).(-7\hat i+2\hat j+3\hat k)$
answered Nov 9, 2013 by rvidyagovindarajan_1

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