Let a unit vector $\overrightarrow{\mathrm{OP}}$ make angle $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $\beta \in\left(0, \frac{\pi}{2}\right)$. $\overrightarrow{\mathrm{OP}}$ 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?

(A) $\alpha \in\left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in\left(\frac{\pi}{2}, \pi\right)$

(B) $\alpha \in\left(0, \frac{\pi}{2}\right)$ and $\gamma \in\left(0, \frac{\pi}{2}\right)$

(C) $\alpha \in\left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in\left(0, \frac{\pi}{2}\right)$

(D) $\alpha \in\left(0, \frac{\pi}{2}\right)$ and $\gamma \in\left(\frac{\pi}{2}, \pi\right)$

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Three Dimensional Geometry

Let a unit vector

OP

make angle

α, β, γ

with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where

β \in (0, \frac{π}{2})

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?

Answer: (A)

The plane through

(1, 2, 3), (2, 3, 4), (1, 5, 7)

has normal

(1, 1, 1) \times (0, 3, 4) = (1, - 4, 3)

, so the unit vector

O P

\pm \frac{1}{26} (1, - 4, 3)

, with direction cosines

cos α = \pm \frac{1}{26}, cos β = \mp \frac{4}{26}, cos γ = \pm \frac{3}{26}

. The condition

β \in (0, \frac{π}{2})

forces

cos β > 0

, which selects

O P = \frac{1}{26} (- 1, 4, - 3)

. Then

cos α = - \frac{1}{26} < 0

and

cos γ = - \frac{3}{26} < 0

, so both

α \in (\frac{π}{2}, π)

and

γ \in (\frac{π}{2}, π) .

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Single correct option