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STD 12 - 11. three dimension geometry — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 11. three dimension geometry4 Marks
MCQ
If a unit vector $\vec r$ makes angles $\frac{\pi }{3}$ with $\hat i$, $\frac{\pi }{4}$ with $\hat j$ and $\theta  \in \left( {0,\pi } \right)$ with  $\hat k$, then a value of $\theta$ is
  • A
    $\frac{{5\pi }}{{12}}$
  • B
    $\frac{{5\pi }}{{6}}$
  • ✓
    $\frac{{2\pi }}{{3}}$
  • D
    $\frac{{\pi }}{{4}}$
✓

Answer

Correct option: C.
$\frac{{2\pi }}{{3}}$
c
$\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$

$\Rightarrow \frac{1}{4}+\frac{1}{2}+\cos ^{2} \gamma=1$

$\Rightarrow \cos ^{2} \gamma=1-\frac{3}{4}=\frac{1}{4}$

$\Rightarrow \cos \gamma=\pm \frac{1}{2}$

$\Rightarrow \gamma=\frac{\pi}{3} \text { or } \frac{2 \pi}{3}$

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