JEE MAIN - Mathematics (2023 - 30th January Morning Shift - No. 9)
Let a unit vector $$\widehat{O P}$$ make angles $$\alpha, \beta, \gamma$$ with the positive directions of the co-ordinate axes $$\mathrm{OX}$$, $$\mathrm{OY}, \mathrm{OZ}$$ respectively, where $$\beta \in\left(0, \frac{\pi}{2}\right)$$. If $$\widehat{\mathrm{OP}}$$ is perpendicular to the plane through points $$(1,2,3),(2,3,4)$$ and $$(1,5,7)$$, then which one of the following is true?
$$\alpha \in\left(\frac{\pi}{2}, \pi\right)$$ and $$\gamma \in\left(\frac{\pi}{2}, \pi\right)$$
$$\alpha \in\left(0, \frac{\pi}{2}\right)$$ and $$\gamma \in\left(\frac{\pi}{2}, \pi\right)$$
$$\alpha \in\left(\frac{\pi}{2}, \pi\right)$$ and $$\gamma \in\left(0, \frac{\pi}{2}\right)$$
$$\alpha \in\left(0, \frac{\pi}{2}\right)$$ and $$\gamma \in\left(0, \frac{\pi}{2}\right)$$
Explanation
Let $$A \equiv (1,2,3),B \equiv (2,3,4),C \equiv (1,5,7)$$
$$\overrightarrow n = \overrightarrow {AB} \times \overrightarrow {AC} = \left| {\matrix{ i & j & k \cr 1 & 1 & 1 \cr 0 & 3 & 4 \cr } } \right|$$
$$ = \widehat i - 4\widehat j + 3\widehat k$$
$$\widehat {OP} = {{ \pm (\widehat i - 4\widehat j + 3\widehat k)} \over {\sqrt {26} }}$$
Since $$\cos \beta > 0$$, take $$-$$ sign
$$\widehat {OP} = {{\widehat i - 4\widehat j + 3\widehat k} \over {\sqrt {26} }}$$
$$ \Rightarrow \cos \alpha < 0,\cos \gamma < 0$$
$$\alpha ,\gamma \in \left( {{\pi \over 2},\pi } \right)$$
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