Mathematics - Vector Question with Solution | TestHub

$\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{a}\right)$
$\Rightarrow -\left(\overrightarrow{a}.\overrightarrow{b}\right)\overrightarrow{c}=\left(\overrightarrow{a}.\overrightarrow{a}\right)\overrightarrow{b}-\left(\overrightarrow{a}.\overrightarrow{b}\right)\overrightarrow{a}$

Using given information $\overrightarrow{a}=\widehat{i}-2\widehat{j}+\widehat{k}$ and $\overrightarrow{b}=\widehat{i}-\widehat{j}+\widehat{k}$ , we can write $\overrightarrow{a}.\overrightarrow{b}=4$, $\overrightarrow{a}.\overrightarrow{a}=6$
$\Rightarrow -4\overrightarrow{c}=6\left(\widehat{i}-\widehat{j}+\widehat{k}\right)-4\left(\widehat{i}-2\widehat{j}+\widehat{k}\right)$
$\Rightarrow -4\overrightarrow{c}=2\widehat{i}+2\widehat{j}+2\widehat{k}$
$\Rightarrow \overrightarrow{c}=-\frac{1}{2}\left(\widehat{i}+\widehat{j}+\widehat{k}\right)$
$\Rightarrow \overrightarrow{b}.\overrightarrow{c}=-\frac{1}{2}$.