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(C) Given that the direction angles are $\alpha = \frac{\pi}{3}$ and $\beta = \frac{\pi}{3}$.Let the angle made by the line with the $Z$-axis be $\gam

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(C) Given that the direction angles are $\alpha = \frac{\pi}{3}$ and $\beta = \frac{\pi}{3}$.Let the angle made by the line with the $Z$-axis be $\gamma$.We know the property of direction cosines: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.Substituting the given values:$\cos^2(\frac{\pi}{3}) + \cos^2(\frac{\pi}{3}) + \cos^2 \gamma = 1$.Since $\cos(\frac{\pi}{3}) = \frac{1}{2}$,we have:$(\frac{1}{2})^2 + (\frac{1}{2})^2 + \cos^2 \gamma = 1$.$\frac{1}{4} + \frac{1}{4} + \cos^2 \gamma = 1$.$\frac{1}{2} + \cos^2 \gamma = 1$.$\cos^2 \gamma = 1 - \frac{1}{2} = \frac{1}{2}$.$\cos \gamma = \pm \frac{1}{\sqrt{2}}$.Since the angle $\gamma$ is acute,$\cos \gamma = \frac{1}{\sqrt{2}}$,which implies $\gamma = \frac{\pi}{4}$.

If a line makes an angle of $\frac{\pi}{3}$ with each $X$ and $Y$ axis,then the acute angle made by $Z$-axis is

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