If a line makes angles of 30^circ and 45^circ | vedclass

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(D) Let the angles made by the line with the $x$,$y$,and $z$ axes be $\alpha$,$\beta$,and $\gamma$ respectively.Given $\alpha = 30^\circ$ and $\beta =

Vedclass · Accepted Answer

None of these

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(D) Let the angles made by the line with the $x$,$y$,and $z$ axes be $\alpha$,$\beta$,and $\gamma$ respectively.Given $\alpha = 30^\circ$ and $\beta = 45^\circ$.The direction cosines of the line are $l = \cos \alpha$,$m = \cos \beta$,and $n = \cos \gamma$.We know that $l^2 + m^2 + n^2 = 1$,which implies $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.Substituting the given values: $\cos^2(30^\circ) + \cos^2(45^\circ) + \cos^2 \gamma = 1$.$(\frac{\sqrt{3}}{2})^2 + (\frac{1}{\sqrt{2}})^2 + \cos^2 \gamma = 1$.$\frac{3}{4} + \frac{1}{2} + \cos^2 \gamma = 1$.$\frac{5}{4} + \cos^2 \gamma = 1$.$\cos^2 \gamma = 1 - \frac{5}{4} = -\frac{1}{4}$.Since the square of a real number cannot be negative,there is no such real angle $\gamma$. Therefore,the correct option is $(d)$.

If a line makes angles of $30^\circ$ and $45^\circ$ with the $x$-axis and $y$-axis respectively,then the angle made by it with the $z$-axis is:

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