Browse Questions

# 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 $(\overrightarrow a+\overrightarrow b).(-7\hat i+2\hat j+3\hat k)=?$

Toolbox:
• 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 $| \overrightarrow a+\overrightarrow b|=\sqrt {29}$,
$\lambda\sqrt {4+9+16}=\sqrt {29}$
$\Rightarrow\:\lambda=1$
$\Rightarrow\:(\overrightarrow a+\overrightarrow b).(-7\hat i+2\hat j+3\hat k)=(2\hat i+3\hat j+4\hat k).(-7\hat i+2\hat j+3\hat k)$
$=-14+6+12=4$