Let a¯=3i¯+2j¯+2k¯ and b¯=i¯+2j¯−2k¯ be two vectors. If a vector perpendicular to both the vectors a¯+b¯ and a¯−b¯ has the magnitude 12. Then one such vector is ______

(a¯+b¯)×(a¯−b¯)=2(b¯×a¯)=2|i¯j¯k¯12−2322|=2(8i¯−8j¯−4k¯)Req vector=12(2i¯−2j¯−K¯)4+4+1=12(2i¯−2j¯−K¯)3=4(2i¯−2j¯−K¯)

Let a¯=3i¯+2j¯+2k¯ and b¯=i¯+2j¯−2k¯ be two vectors. If a vector perpendicular to both the vectors a¯+b¯ and a¯−b¯ has the magnitude 12. Then one such vector is ______

(a¯+b¯)×(a¯−b¯)=2(b¯×a¯)=2|i¯j¯k¯12−2322|=2(8i¯−8j¯−4k¯)Req vector=12(2i¯−2j¯−K¯)4+4+1=12(2i¯−2j¯−K¯)3=4(2i¯−2j¯−K¯)

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

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$4 (- 2 \bar{i} - 2 \bar{j} + \bar{k})$

$4 (2 \bar{i} + 2 \bar{j} + \bar{k})$

$4 (2 \bar{i} + 2 \bar{j} - \bar{k})$

$4 (2 \bar{i} - 2 \bar{j} - \bar{k})$

answer is C.

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Detailed Solution

$(\bar{a} + \bar{b}) \times (\bar{a} - \bar{b}) = 2 (\bar{b} \times \bar{a}) = 2 | \begin{array}{ccc} \bar{i} & \bar{j} & \bar{k} \\ 1 & 2 & - 2 \\ 3 & 2 & 2 \end{array} | = 2 (8 \bar{i} - 8 \bar{j} - 4 \bar{k})$

$Req vector = \frac{12 (2 \bar{i} - 2 \bar{j} - \bar{K})}{\sqrt{4 + 4 + 1}} = \frac{12 (2 \bar{i} - 2 \bar{j} - \bar{K})}{3} = 4 (2 \bar{i} - 2 \bar{j} - \bar{K})$