Verified Solution • 2025–26 Session moderate

QJEEM19Ma398a: JEE Main 2019 Physics - Class 11 JEEM19Ma398a Solution: If a unit vector makes angles /3 with , / 4 with and (0, ) with …

If a unit vector a→\overrightarrow a makes angles π\pi /3 with i^\widehat i , π\pi / 4 with j^\widehat j and θ\theta ∈\in (0, π\pi ) with k^\widehat k , then a value of θ\theta is :-

Multiple Choice Options

  • A 5π6{{5\pi } \over {6}}
  • B 5π12{{5\pi } \over {12}}
  • C 2π3{{2\pi } \over {3}}
  • D π4{{\pi } \over {4}}

Step-by-Step Verified Solution

A unit vector a→\overrightarrow a makes angles π\pi /3 with i^\widehat i

∴\therefore α\alpha = π\pi /3

and π\pi / 4 with j^\widehat j

∴\therefore β\beta = π\pi / 4

and θ\theta ∈\in (0, π\pi ) with k^\widehat k

∴\therefore γ\gamma = θ\theta

We also know,

cos⁡2α+cos⁡2β+cos⁡2γ{\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1

⇒\Rightarrow cos⁡2π3+cos⁡2π4+cos⁡2θ{\cos ^2}{\pi \over 3} + {\cos ^2}{\pi \over 4} + {\cos ^2}\theta = 1

⇒\Rightarrow 14+12+cos⁡2θ{1 \over 4} + {1 \over 2} + {\cos ^2}\theta = 1

⇒\Rightarrow cos⁡2θ{\cos ^2}\theta = ±12\pm {1 \over 2}

⇒\Rightarrow θ\theta = π3{\pi \over 3} or 2π3{{2\pi } \over {3}}

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