JEE MAIN - Mathematics (2021 - 25th July Morning Shift - No. 17)
Let $$\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$$ and $$\overrightarrow q = \widehat i + 2\widehat j + \widehat k$$ be two vectors. If a vector $$\overrightarrow r = (\alpha \widehat i + \beta \widehat j + \gamma \widehat k)$$ is perpendicular to each of the vectors ($$(\overrightarrow p + \overrightarrow q )$$ and $$(\overrightarrow p - \overrightarrow q )$$, and $$\left| {\overrightarrow r } \right| = \sqrt 3 $$, then $$\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|$$ is equal to _______________.
Answer
3
Explanation
$$\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$$ (Given )
$$\overrightarrow q = \widehat i + 2\widehat j + \widehat k$$
Now, $$(\overrightarrow p + \overrightarrow q ) \times (\overrightarrow p - \overrightarrow q ) = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 3 & 5 & 2 \cr 1 & 1 & 0 \cr } } \right|$$
$$ = - 2\widehat i - 2\widehat j - 2\widehat k$$
$$ \Rightarrow \overrightarrow r = \pm \sqrt 3 {{\left( {(\overrightarrow p + \overrightarrow q ) \times (\overrightarrow p - \overrightarrow q )} \right)} \over {\left| {(\overrightarrow p + \overrightarrow q ) \times (\overrightarrow p - \overrightarrow q )} \right|}} = \pm {{\sqrt 3 \left( { - 2\widehat i - 2\widehat j - 2\widehat k} \right)} \over {\sqrt {{2^2} + {2^2} + {2^2}} }}$$
$$\overrightarrow r = \pm \left( { - \widehat i - \widehat j - \widehat k} \right)$$
According to question
$$\overrightarrow r = \alpha \widehat i + \beta \widehat j + \gamma \widehat k$$
So, |$$\alpha$$| = 1, |$$\beta$$| = 1, |$$\gamma$$| = 1
$$\Rightarrow$$ $$\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|$$ = 3
$$\overrightarrow q = \widehat i + 2\widehat j + \widehat k$$
Now, $$(\overrightarrow p + \overrightarrow q ) \times (\overrightarrow p - \overrightarrow q ) = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 3 & 5 & 2 \cr 1 & 1 & 0 \cr } } \right|$$
$$ = - 2\widehat i - 2\widehat j - 2\widehat k$$
$$ \Rightarrow \overrightarrow r = \pm \sqrt 3 {{\left( {(\overrightarrow p + \overrightarrow q ) \times (\overrightarrow p - \overrightarrow q )} \right)} \over {\left| {(\overrightarrow p + \overrightarrow q ) \times (\overrightarrow p - \overrightarrow q )} \right|}} = \pm {{\sqrt 3 \left( { - 2\widehat i - 2\widehat j - 2\widehat k} \right)} \over {\sqrt {{2^2} + {2^2} + {2^2}} }}$$
$$\overrightarrow r = \pm \left( { - \widehat i - \widehat j - \widehat k} \right)$$
According to question
$$\overrightarrow r = \alpha \widehat i + \beta \widehat j + \gamma \widehat k$$
So, |$$\alpha$$| = 1, |$$\beta$$| = 1, |$$\gamma$$| = 1
$$\Rightarrow$$ $$\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|$$ = 3
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