Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Let the unit vector be $\vec{a}=\cos \alpha \hat{i}+\cos \beta \hat{j}+\cos \gamma \hat{k}$.

Given $\alpha =\frac{\pi }{4}$, $\beta =\frac{\pi }{3}$, and $\gamma =\theta$.

Using the property of direction cosines, ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$.

Substituting the given values:

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

${\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2+{\cos }^2\theta =1$

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

$\frac{3}{4}+{\cos }^2\theta =1$

${\cos }^2\theta =\frac{1}{4}$

$\cos \theta =\pm \frac{1}{2}$

Since $\theta \in (0,\pi )$, we get $\theta =\frac{\pi }{3}$ or $\theta =\frac{2\pi }{3}$.

Answer: $\frac{\pi }{3},\frac{2\pi }{3}$