A line makes an angle of 45^{circ} with the x | vedclass

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Question

(A) Let the angles made by the line with the $x$,$y$,and $z$-axes be $\alpha$,$\beta$,and $\gamma$ respectively. Given $\alpha = 45^{\circ}$ and $\bet

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$ and $-\frac{1}{\sqrt{2}},-\frac{1}{2},-\frac{1}{2}$

Vedclass · Answer

(A) Let the angles made by the line with the $x$,$y$,and $z$-axes be $\alpha$,$\beta$,and $\gamma$ respectively. Given $\alpha = 45^{\circ}$ and $\beta = \gamma = 	heta$. The direction cosines are $l = \cos \alpha$,$m = \cos \beta$,and $n = \cos \gamma$. We know that $l^2 + m^2 + n^2 = 1$. Substituting the values,we get $\cos^2 45^{\circ} + \cos^2 	heta + \cos^2 	heta = 1$. $(\frac{1}{\sqrt{2}})^2 + 2 \cos^2 	heta = 1$. $\frac{1}{2} + 2 \cos^2 	heta = 1$. $2 \cos^2 	heta = 1 - \frac{1}{2} = \frac{1}{2}$. $\cos^2 	heta = \frac{1}{4} \Rightarrow \cos 	heta = \pm \frac{1}{2}$. Thus,the direction cosines are $(\cos 45^{\circ}, \cos 	heta, \cos 	heta)$,which gives $(\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2})$ or $(\frac{1}{\sqrt{2}}, -\frac{1}{2}, -\frac{1}{2})$.

$A$ line makes an angle of $45^{\circ}$ with the $x$-axis and congruent angles with the $y$ and $z$-axes. Then,the direction cosines of the line are:

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