The direction cosines of the line making angl | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the angles made by the line with the $X, Y$ and $Z$ axes be $\alpha = \frac{\pi}{4}$,$\beta = \frac{\pi}{3}$,and $\gamma = 	heta$. The direct

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) Let the angles made by the line with the $X, Y$ and $Z$ axes be $\alpha = \frac{\pi}{4}$,$\beta = \frac{\pi}{3}$,and $\gamma = 	heta$. The direction cosines are given by $l = \cos \alpha$,$m = \cos \beta$,and $n = \cos \gamma$. We know that $l^2 + m^2 + n^2 = 1$. Substituting the values: $(\cos \frac{\pi}{4})^2 + (\cos \frac{\pi}{3})^2 + \cos^2 	heta = 1$. $(\frac{1}{\sqrt{2}})^2 + (\frac{1}{2})^2 + \cos^2 	heta = 1$. $\frac{1}{2} + \frac{1}{4} + \cos^2 	heta = 1$. $\frac{3}{4} + \cos^2 	heta = 1$. $\cos^2 	heta = 1 - \frac{3}{4} = \frac{1}{4}$. Since $0 Thus,the direction cosines are $\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}$,$\cos \frac{\pi}{3} = \frac{1}{2}$,and $\cos 	heta = \frac{1}{2}$. Therefore,the direction cosines are $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$.

The direction cosines of the line making angles $\frac{\pi}{4}, \frac{\pi}{3}$ and $\theta$ $(0 < \theta < \frac{\pi}{2})$ respectively with $X, Y$ and $Z$ axes are:

Similar Questions

If the direction cosines of a line are $\left( \frac{1}{c}, \frac{1}{c}, \frac{1}{c} \right)$,then

If the projections of a line on the coordinate axes are $4, 6, 12$,what are the direction cosines of the line?

If a line makes an angle of $\frac{\pi}{4}$ with the positive directions of each of the $x$-axis and $y$-axis,then the angle that the line makes with the positive direction of the $z$-axis is

If $l, m, n$ are the direction cosines of a line that is perpendicular to the lines having the direction ratios $1, 2, -1$ and $1, -2, 1$,then $(l+m+n)^2=$

If $\alpha, \beta$ and $\gamma$ are the angles which a half ray makes with the positive direction of the axes,then $\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module