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
Correct Option: , 3
$(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})$
$=2(\vec{b} \times \vec{a})$
$=2\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 1 & 2 & -2 \\ 3 & 2 & 2\end{array}\right|$
$=2(8 \hat{\mathrm{i}}-8 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})$Required vector $=\pm 12 \frac{(2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-\hat{\mathrm{k}})}{3}$
$=\pm 4(2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-\hat{\mathrm{k}})$
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