JEE MAIN - Mathematics (2019 - 12th April Morning Slot - No. 14)
Let $$\overrightarrow a = 3\widehat i + 2\widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j - 2\widehat k$$ be two vectors. If a vector perpendicular to both the vectors
$$\overrightarrow a + \overrightarrow b $$ and $$\overrightarrow a - \overrightarrow b $$ has the magnitude 12 then one such vector is :
$$4\left( {2\widehat i - 2\widehat j - \widehat k} \right)$$
$$4\left( { - 2\widehat i - 2\widehat j + \widehat k} \right)$$
$$4\left( {2\widehat i + 2\widehat j + \widehat k} \right)$$
$$4\left( {2\widehat i + 2\widehat j - \widehat k} \right)$$
Explanation
Required vector is $\overrightarrow r$ = $$\lambda \left( {\left( {\overline a + \overline b } \right) \times \left( {\overline a - \overline b } \right)} \right)$$
$$ \Rightarrow \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 4 & 4 & 0 \cr 2 & 0 & 4 \cr } } \right| = \lambda \left( {16\widehat i - 16\widehat j - 8\widehat k} \right)$$
$$ \Rightarrow \overrightarrow r = 8\lambda \left( {2\widehat i - 2\widehat j - \widehat k} \right) = \left| {\overrightarrow r } \right|$$
$$ \Rightarrow \left| {8\lambda } \right|.3 \Rightarrow 8\lambda = \pm 4$$
$$ \Rightarrow \overrightarrow r = \pm 4\left( {2\widehat i - 2\widehat j - \widehat k} \right)$$
$$ \Rightarrow \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 4 & 4 & 0 \cr 2 & 0 & 4 \cr } } \right| = \lambda \left( {16\widehat i - 16\widehat j - 8\widehat k} \right)$$
$$ \Rightarrow \overrightarrow r = 8\lambda \left( {2\widehat i - 2\widehat j - \widehat k} \right) = \left| {\overrightarrow r } \right|$$
$$ \Rightarrow \left| {8\lambda } \right|.3 \Rightarrow 8\lambda = \pm 4$$
$$ \Rightarrow \overrightarrow r = \pm 4\left( {2\widehat i - 2\widehat j - \widehat k} \right)$$
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