(D) Let the direction cosines of the line be $l, m, n$. Since the line makes equal angles $\alpha$ with the coordinate axes,we have $l = \cos \alpha$,

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(D) Let the direction cosines of the line be $l, m, n$. Since the line makes equal angles $\alpha$ with the coordinate axes,we have $l = \cos \alpha$,$m = \cos \alpha$,and $n = \cos \alpha$. We know that for any line,$l^2 + m^2 + n^2 = 1$. Substituting the values,we get $\cos^2 \alpha + \cos^2 \alpha + \cos^2 \alpha = 1$. $3 \cos^2 \alpha = 1$. $\cos^2 \alpha = 1/3$. $\cos \alpha = 1/\sqrt{3}$ (taking the positive value for the acute angle). Therefore,$\alpha = \cos^{-1}(1/\sqrt{3})$.

Class 12 Mathematics THREE DIMENSIONAL GEOMETRY System of co-ordinates, Direction cosines and direction ratios, Projection

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If a line in a plane makes equal angles with the coordinate axes,then the angle made by the line with any one axis is equal to .........

A
$60^\circ$
B
$45^\circ$
C
$\cos^{-1}(1/3)$
D
$\cos^{-1}(1/\sqrt{3})$

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THREE DIMENSIONAL GEOMETRY

Subtopic

System of co-ordinates, Direction cosines and direction ratios, Projection

If a line makes angles $90^{\circ}, 60^{\circ}$ and $30^{\circ}$ with the positive direction of $x, y$ and $z$-axes respectively,find its direction cosines.

Easy

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If a line makes angles $\alpha, \beta, \gamma$ with the three-dimensional coordinate axes respectively,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = $

Easy

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Let the direction cosines of two lines satisfy the equations $3l+2m+n=0$ and $2mn-3nl+5lm=0$. If $\theta$ is the angle between these two lines,then $\cos \theta=$

EasyTS EAMCET 2023

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If a line in space makes angles $\alpha, \beta$,and $\gamma$ with the coordinate axes,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma + \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ equals:

MediumAP EAMCET 2009

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If $l_1, m_1, n_1$; $l_2, m_2, n_2$ and $l_3, m_3, n_3$ are the direction cosines of three mutually perpendicular lines,find the direction cosines of a line that makes equal angles with these lines.

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If a line makes angles $90^{\circ}, 60^{\circ}$ and $30^{\circ}$ with the positive direction of $x, y$ and $z$-axes respectively,find its direction cosines.

If a line makes angles $\alpha, \beta, \gamma$ with the three-dimensional coordinate axes respectively,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = $

Let the direction cosines of two lines satisfy the equations $3l+2m+n=0$ and $2mn-3nl+5lm=0$. If $\theta$ is the angle between these two lines,then $\cos \theta=$

If a line in space makes angles $\alpha, \beta$,and $\gamma$ with the coordinate axes,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma + \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ equals:

If $l_1, m_1, n_1$; $l_2, m_2, n_2$ and $l_3, m_3, n_3$ are the direction cosines of three mutually perpendicular lines,find the direction cosines of a line that makes equal angles with these lines.

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