# If $\overrightarrow a$ and $\overrightarrow b$ are such that $|\overrightarrow a+\overrightarrow b|=\sqrt {29}$ 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 the possible value of $(\overrightarrow a+\overrightarrow b).(-7\hat i+2\hat j+3\hat k)=?$

$\begin{array}{1 1} 0 \\ 3 \\ 4 \\ 8 \end{array}$

Toolbox:
• $\overrightarrow a\times\overrightarrow b=-\overrightarrow b\times\overrightarrow a$
Given: $\overrightarrow a\times (2\hat i+3\hat j+4\hat k)=(2\hat i+3\hat j+4\hat k)\times \overrightarrow b$
and $|\overrightarrow a+\overrightarrow b|=\sqrt {29}$
$\Rightarrow\:(\overrightarrow a+\overrightarrow b)\times (2\hat i+3\hat j+4\hat k)=0$
$\Rightarrow\:(\overrightarrow a+\overrightarrow b)$ is parallel to $(2\hat i+3\hat j+4\hat k)$
and hence
$\overrightarrow a+\overrightarrow b=\large\frac{2\hat i+3\hat j+4\hat k}{\sqrt {29}}.$$\sqrt {29}=2\hat i+3\hat j+4\hat k$
Now
$(\overrightarrow a+\overrightarrow b).(-7\hat i+2\hat j+3\hat k)$
$=-14+6+12=4$