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

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(B) Let the angles made by the line with the $x, y,$ and $z$ axes be $\alpha, \beta,$ and $\gamma$ respectively.Given $\alpha = 120^{\circ}$ and $\bet

Vedclass · Accepted Answer

$45^{\circ}$ or $135^{\circ}$

Vedclass · Answer

(B) Let the angles made by the line with the $x, y,$ and $z$ axes be $\alpha, \beta,$ and $\gamma$ respectively.Given $\alpha = 120^{\circ}$ and $\beta = 60^{\circ}$.We know that the sum of the squares of the direction cosines is $1$,i.e.,$\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$.Substituting the given values:$\cos^{2} (120^{\circ}) + \cos^{2} (60^{\circ}) + \cos^{2} \gamma = 1$Since $\cos(120^{\circ}) = -\frac{1}{2}$ and $\cos(60^{\circ}) = \frac{1}{2}$,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}}$Therefore,$\gamma = 45^{\circ}$ or $135^{\circ}$.

If a line makes angles of $120^{\circ}$ and $60^{\circ}$ with the $x$ and $y$ axes respectively,what angle does it make with the $z$ axis?

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