Question

Let $\vec{a} = 3 \hat{i} + 2\hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - 2 \hat{k} $ be two vectors. If a vector perpendicular to both the vectors $\vec{a} + \vec{b}$ and $ \vec{a } - \vec{b}$ has the magnitude $12$ then one such vector is

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

Answer 1

Answer: (C)

$\left(\vec{a} + \vec{b}\right) \times\left(\vec{a} - \vec{b}\right) $
$ = 2 \left(\vec{b} \times\vec{a}\right) $
$ = 2 \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 1&2&-2\\ 3&2&2\end{vmatrix} $
$ = 2 \left(8\hat{i} -8\hat{j} +4\hat{k}\right) $
Required vector $ = \pm 12 \frac{\left(2\hat{i} -2 \hat{j} -\hat{k}\right)}{3} $
$ = \pm 4 \left(2 \hat{i} - 2\hat{j} -\hat{k}\right) $