If the line overrightarrow{OR} makes angles t | vedclass

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

Question

(B) Let the direction cosines of the line $\overrightarrow{OR}$ be $l, m, n$. Since the line makes angles $	heta_{1}, 	heta_{2}, 	heta_{3}$ with th

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) Let the direction cosines of the line $\overrightarrow{OR}$ be $l, m, n$. Since the line makes angles $	heta_{1}, 	heta_{2}, 	heta_{3}$ with the planes $XOY, YOZ, ZOX$ respectively,the angles with the normals to these planes (which are the $Z, X, Y$ axes) are $\frac{\pi}{2}-	heta_{1}, \frac{\pi}{2}-	heta_{2}, \frac{\pi}{2}-	heta_{3}$. Thus,$|l| = \cos(\frac{\pi}{2}-	heta_{2}) = \sin 	heta_{2}$,$|m| = \cos(\frac{\pi}{2}-	heta_{3}) = \sin 	heta_{3}$,and $|n| = \cos(\frac{\pi}{2}-	heta_{1}) = \sin 	heta_{1}$. We know that for direction cosines,$l^{2}+m^{2}+n^{2}=1$. Substituting the values,we get $\sin^{2} 	heta_{2} + \sin^{2} 	heta_{3} + \sin^{2} 	heta_{1} = 1$. Using the identity $\sin^{2} 	heta = 1 - \cos^{2} 	heta$,we have $(1-\cos^{2} 	heta_{2}) + (1-\cos^{2} 	heta_{3}) + (1-\cos^{2} 	heta_{1}) = 1$. $3 - (\cos^{2} 	heta_{1} + \cos^{2} 	heta_{2} + \cos^{2} 	heta_{3}) = 1$. Therefore,$\cos^{2} 	heta_{1} + \cos^{2} 	heta_{2} + \cos^{2} 	heta_{3} = 3 - 1 = 2$.

If the line $\overrightarrow{OR}$ makes angles $\theta_{1}, \theta_{2}, \theta_{3}$ with the planes $XOY, YOZ, ZOX$ respectively,then $\cos ^{2} \theta_{1}+\cos ^{2} \theta_{2}+\cos ^{2} \theta_{3}$ is equal to

Similar Questions

The angle between a line with direction ratios $2, 2, 1$ and a line joining $(3, 1, 4)$ and $(7, 2, 12)$ is

The direction cosines of two lines are connected by the relations $l-m+n=0$ and $2lm-3mn+nl=0$. If $\theta$ is the angle between these two lines, then $\cos \theta=$

If the projections of a line segment on the coordinate axes are $2, 3, 6$,find the length of the line segment.

If the direction ratios of two lines are $5, -12, 13$ and $-3, 4, 5$,then the angle between them is:

If a line makes angles of measure $\frac{\pi}{6}$ and $\frac{\pi}{3}$ with $X$ and $Y$ axes respectively,then the angle made by the line with $Z$ axis is

Vedclass Test Series

Exam Paper Generator

Online Exam Module