Verified Solution • 2025–26 Session moderate

QJEEM23Maa36b: JEE Main 2023 Physics - Class 11 JEEM23Maa36b Solution: Let a unit vector make angles with the positive directions of…

Let a unit vector OP^\widehat{O P} make angles α,β,γ\alpha, \beta, \gamma with the positive directions of the co-ordinate axes OX\mathrm{OX} , OY,OZ\mathrm{OY}, \mathrm{OZ} respectively, where β∈(0,π2)\beta \in\left(0, \frac{\pi}{2}\right) . If OP^\widehat{\mathrm{OP}} is perpendicular to the plane through points (1,2,3),(2,3,4)(1,2,3),(2,3,4) and (1,5,7)(1,5,7) , then which one of the following is true?

Multiple Choice Options

  • A α∈(π2,π)\alpha \in\left(\frac{\pi}{2}, \pi\right) and γ∈(π2,π)\gamma \in\left(\frac{\pi}{2}, \pi\right)
  • B α∈(0,π2)\alpha \in\left(0, \frac{\pi}{2}\right) and γ∈(π2,π)\gamma \in\left(\frac{\pi}{2}, \pi\right)
  • C α∈(π2,π)\alpha \in\left(\frac{\pi}{2}, \pi\right) and γ∈(0,π2)\gamma \in\left(0, \frac{\pi}{2}\right)
  • D α∈(0,π2)\alpha \in\left(0, \frac{\pi}{2}\right) and γ∈(0,π2)\gamma \in\left(0, \frac{\pi}{2}\right)

Step-by-Step Verified Solution

Let A≡(1,2,3),B≡(2,3,4),C≡(1,5,7)A \equiv (1,2,3),B \equiv (2,3,4),C \equiv (1,5,7)

n→=AB→×AC→=|ijk111034|

=i^−4j^+3k^= \widehat i - 4\widehat j + 3\widehat k

OP^=±(i^−4j^+3k^)26\widehat {OP} = {{ \pm (\widehat i - 4\widehat j + 3\widehat k)} \over {\sqrt {26} }}

Since \cos \beta > 0 , take −- sign

OP^=i^−4j^+3k^26\widehat {OP} = {{\widehat i - 4\widehat j + 3\widehat k} \over {\sqrt {26} }}

\Rightarrow \cos \alpha < 0,\cos \gamma < 0

α,γ∈(π2,π)\alpha ,\gamma \in \left( {{\pi \over 2},\pi } \right)

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