Question

Find $\overrightarrow{a}.\left(\overrightarrow{b}\times \overrightarrow{c}\right)$, if $\overrightarrow{a}=2\hat{i}+\hat{j}+3\hat{k},\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}+2\hat{k}$. [CBSE 2014]

Answer 1

The given vectors are $\overrightarrow{a}=2\hat{i}+\hat{j}+3\hat{k},\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}+2\hat{k}$.

Now,

$\overrightarrow{b}\times \overrightarrow{c}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -1& 2& 1\\ 3& 1& 2\end{array}\right|=3\hat{i}+5\hat{j}-7\hat{k}$

$\therefore \overrightarrow{a}.\left(\overrightarrow{b}\times \overrightarrow{c}\right)=\left(2\hat{i}+\hat{j}+3\hat{k}\right).\left(3\hat{i}+5\hat{j}-7\hat{k}\right)=2\times 3+1\times 5+3\times \left(-7\right)=6+5-21=-10$