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Need clarity, kindly explain! - Three Dimensional Geometry - JEE Main-6

If a unit vector  \vec{a} make angles \pi/3    with  \hat{i},\pi/4\:\:\:with\:\:\:\hat{j}\:\:and\:\:\theta\:\epsilon (0,\pi)\:\:with\:\:\hat{k} then a value of \theta  is :

  • Option 1)

    \frac{5\pi}{6}

  • Option 2)

    \frac{\pi}{4}

  • Option 3)

  • Option 4)

    \frac{2\pi}{3}

 
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\\\cos^{2}\frac{\pi}{3}+\cos^{2}\frac{\pi}{4}+\cos^{2}\theta =1\\\\\\\:\left ( \frac{1}{2} \right )^{2}+\left ( \frac{1}{\sqrt{2}} \right )^{2}+\cos^{2}\theta=1\\\\\\\:\frac{1}{4}+\frac{1}{2}+\cos^{2}\theta=1

\\\cos^{2}\theta=\frac{1}{4}\\\\\\\:\cos\theta=\pm \frac{1}{2}\\\\\\\:\therefore \frac{\pi}{3}\:\:\:or\:\:\:\pi-\frac{\pi}{3}=\frac{2\pi}{3}

 


Option 1)

\frac{5\pi}{6}

Option 2)

\frac{\pi}{4}

Option 3)

Option 4)

\frac{2\pi}{3}

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