Question

If $\overrightarrow{a}=2\hat{i}+\hat{j}+2\hat{k}$ , then the value of $\hat{i}\times {(\overrightarrow{a}\times \hat{i})}^2+\hat{j}\times {(\overrightarrow{a}\times \hat{j})}^2+\hat{k}\times {(\overrightarrow{a}\times \hat{k})}^2$is equal to

Answer 1

Solution:

On simplifying the vectors, we get,

$\begin{array}{rcl}|\hat{i}\times (\overrightarrow{a}\times \hat{i}){|}^2& =& {\left|\overrightarrow{a}–(a.\hat{i})\hat{i}\right|}^2\\ & =& {\left|\hat{j}+2\hat{k}\right|}^2\\ & =& 1+4\\ & =& 5\end{array}$

Similarly,

$\begin{array}{rcl}{\left|\hat{j}\times (\overrightarrow{a}\times \hat{j})\right|}^2& =& {\left|2\hat{i}+2\hat{k}\right|}^2\\ & =& 4+4\\ & =& 8\end{array}$

Again,

$\begin{array}{rcl}{\left|\hat{k}\times (\overrightarrow{a}\times \widehat{k)}\right|}^2& =& {\left|2\hat{i}+\hat{j}\right|}^2\\ & =& 4+1\\ & =& 5\end{array}$

On addition of these three equations, we get,

$\begin{array}{rcl}\hat{i}\times {(\overrightarrow{a}\times \hat{i})}^2+\hat{j}\times {(\overrightarrow{a}\times \hat{j})}^2+\hat{k}\times {(\overrightarrow{a}\times \hat{k})}^2& =& 5+8+5\\ & =& 18\end{array}$

Hence, the required answer is 18.