A line makes angles of 45^circ and 60^circ wi | vedclass

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(D) Let the angles made by the line with the positive $X$,$Y$,and $Z$ axes be $\alpha$,$\beta$,and $\gamma$ respectively.Given $\alpha = 45^\circ$ and

Vedclass · Accepted Answer

$60^\circ$ or $120^\circ$

Vedclass · Answer

(D) Let the angles made by the line with the positive $X$,$Y$,and $Z$ axes be $\alpha$,$\beta$,and $\gamma$ respectively.Given $\alpha = 45^\circ$ and $\beta = 60^\circ$.The direction cosines of the line are $l = \cos \alpha$,$m = \cos \beta$,and $n = \cos \gamma$.We know that $l^2 + m^2 + n^2 = 1$.Substituting the values: $\cos^2 45^\circ + \cos^2 60^\circ + \cos^2 \gamma = 1$.$\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + \cos^2 \gamma = 1$.$\frac{1}{2} + \frac{1}{4} + \cos^2 \gamma = 1$.$\frac{3}{4} + \cos^2 \gamma = 1$.$\cos^2 \gamma = 1 - \frac{3}{4} = \frac{1}{4}$.$\cos \gamma = \pm \frac{1}{2}$.Thus,$\gamma = 60^\circ$ or $120^\circ$.

$A$ line makes angles of $45^\circ$ and $60^\circ$ with the positive axes of $X$ and $Y$ respectively. The angle made by the same line with the positive axis of $Z$ is:

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