The direction cosines of a line which lies in | vedclass

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

Question

(B) Direction cosines are the cosines of the angles which a line makes with the positive coordinate axes. They are represented by $\langle l, m, n \ra

Vedclass · Accepted Answer

$\pm \frac{1}{2}, 0, \frac{\sqrt{3}}{2}$

Vedclass · Answer

(B) Direction cosines are the cosines of the angles which a line makes with the positive coordinate axes. They are represented by $\langle l, m, n \rangle$ where $l, m, n$ correspond to the $x$-axis,$y$-axis,and $z$-axis respectively. The sum of the squares of the direction cosines is unity,i.e.,$l^{2} + m^{2} + n^{2} = 1$.Since the line lies in the $ZOX$ plane,it is perpendicular to the $y$-axis. Therefore,the angle made with the $y$-axis is $90^{\circ}$,so $m = \cos(90^{\circ}) = 0$.The line makes an angle of $30^{\circ}$ with the $Z$-axis,so $n = \cos(30^{\circ}) = \frac{\sqrt{3}}{2}$.Using the property $l^{2} + m^{2} + n^{2} = 1$,we substitute $m = 0$ and $n = \frac{\sqrt{3}}{2}$:$l^{2} + 0^{2} + \left(\frac{\sqrt{3}}{2}\right)^{2} = 1$$l^{2} + \frac{3}{4} = 1$$l^{2} = 1 - \frac{3}{4} = \frac{1}{4}$$l = \pm \frac{1}{2}$.Thus,the direction cosines are $\langle \pm \frac{1}{2}, 0, \frac{\sqrt{3}}{2} \rangle$.

The direction cosines of a line which lies in the $ZOX$ plane and makes an angle of $30^{\circ}$ with the $Z$-axis are

Similar Questions

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:

Find the position vector of a point $A$ in space such that $\overrightarrow{OA}$ is inclined at $60^{\circ}$ to $OX$ and at $45^{\circ}$ to $OY$ and $|\overrightarrow{OA}|=10$ units.

$A$ line passes through the points $A(6, -7, -1)$ and $B(2, -3, 1)$. Find the direction cosines of the line such that the angle made by the line with the positive direction of the $x$-axis is acute.

$A$ line makes angles $\alpha, \beta, \gamma$ with the coordinate axes and $\alpha+\beta=90^{\circ}$,then $\gamma=$ (in $^{\circ}$)

Suppose $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ are the direction cosines of two lines and $\theta$ is the angle between them,where $\cos \theta = \pm(l_1 l_2 + m_1 m_2 + n_1 n_2)$. Let $A=(1, -2, 3)$,$B=(3, 1, -3)$,and $C=(-3, 1, 3)$ be the vertices of $\triangle ABC$. Then,$\cos A =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module