If a line L makes angles frac{pi}{3} and frac | vedclass

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(B) Let the angles made by the line $L$ with the positive $X$,$Y$,and $Z$ axes be $\alpha$,$\beta$,and $\gamma$ respectively. Given that $\alpha = \fr

Vedclass · Accepted Answer

$\frac{\pi}{3}$

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(B) Let the angles made by the line $L$ with the positive $X$,$Y$,and $Z$ axes be $\alpha$,$\beta$,and $\gamma$ respectively. Given that $\alpha = \frac{\pi}{3}$ and $\beta = \frac{\pi}{4}$. The relationship between direction cosines is given by $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Substituting the given values: $\cos^2 \frac{\pi}{3} + \cos^2 \frac{\pi}{4} + \cos^2 \gamma = 1$ $\Rightarrow (\frac{1}{2})^2 + (\frac{1}{\sqrt{2}})^2 + \cos^2 \gamma = 1$ $\Rightarrow \frac{1}{4} + \frac{1}{2} + \cos^2 \gamma = 1$ $\Rightarrow \frac{3}{4} + \cos^2 \gamma = 1$ $\Rightarrow \cos^2 \gamma = 1 - \frac{3}{4} = \frac{1}{4}$ $\Rightarrow \cos \gamma = \pm \frac{1}{2}$. Since the question asks for the angle with the positive direction of the $Z$-axis,we take $\cos \gamma = \frac{1}{2}$. Therefore,$\gamma = \frac{\pi}{3}$.

If a line $L$ makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the positive $X$-axis and positive $Y$-axis respectively,then the angle made by $L$ with the positive direction of $Z$-axis is

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