Question

If $\overrightarrow{a}=3\hat{i}+2\hat{j}+2\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}-2\hat{k}$. Then a vector of magnitude $12$, which is perpendicular to both $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$, is?

• A. $4(2\hat{i}+2\hat{j}+\hat{k})$
• B. $4(2\hat{i}-2\hat{j}-\hat{k})$
• C. $4(\hat{i}-2\hat{j}-2\hat{k})$
• D. $4(2\hat{i}+\hat{j}+2\hat{k})$

Answer 1

The correct option is B $4(2\hat{i}-2\hat{j}-\hat{k})$

Required vector is $\overrightarrow{r}=\lambda \left(\right(\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}-\overrightarrow{b}\left)\right)=\lambda \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 4& 4& 0\\ 2& 0& 4\end{array}\right|=\lambda (16\hat{i}-16\hat{j}-8\hat{k})$
$\Rightarrow r=8\lambda (2\hat{i}-2\hat{j}-\hat{k})$
$\Rightarrow |\overrightarrow{r}|=|8\lambda |\cdot 3$
$\Rightarrow 8\lambda =\pm 4$
$\overrightarrow{r}=\pm 4(2\hat{i}-2\hat{k}-\hat{k})$.