If a line makes angles 120^{circ} and 60^{cir | vedclass

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Question

(C) Let the direction angles be $\alpha, \beta, \gamma$ with $X, Y, Z$ axes respectively.Given $\alpha = 120^{\circ}$ and $\gamma = 60^{\circ}$.We kno

Vedclass · Accepted Answer

$135$

Vedclass · Answer

(C) Let the direction angles be $\alpha, \beta, \gamma$ with $X, Y, Z$ axes respectively.Given $\alpha = 120^{\circ}$ and $\gamma = 60^{\circ}$.We know that $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.Substituting the values: $\cos^2(120^{\circ}) + \cos^2 \beta + \cos^2(60^{\circ}) = 1$.$(-\frac{1}{2})^2 + \cos^2 \beta + (\frac{1}{2})^2 = 1$.$\frac{1}{4} + \cos^2 \beta + \frac{1}{4} = 1$.$\cos^2 \beta = 1 - \frac{1}{2} = \frac{1}{2}$.$\cos \beta = \pm \frac{1}{\sqrt{2}}$.Thus,$\beta = 45^{\circ}$ or $\beta = 135^{\circ}$.Since $135^{\circ}$ is one of the options,the correct answer is $135^{\circ}$.

If a line makes angles $120^{\circ}$ and $60^{\circ}$ with the positive directions of $X$ and $Z$ axes respectively,then the angle made by the line with the positive $Y$-axis is (in $^{\circ}$)

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