The direction cosines of a line which makes e | vedclass

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

Question

(C) Let the direction angles of the line be $\alpha, \beta, \gamma$. Since the line makes equal angles with the coordinate axes,we have $\alpha = \bet

Vedclass · Accepted Answer

$\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

Vedclass · Answer

(C) Let the direction angles of the line be $\alpha, \beta, \gamma$. Since the line makes equal angles with the coordinate axes,we have $\alpha = \beta = \gamma$. The direction cosines are given by $l = \cos \alpha, m = \cos \beta, n = \cos \gamma$. We know the identity $l^2 + m^2 + n^2 = 1$,which implies $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Substituting $\alpha = \beta = \gamma$,we get $3 \cos^2 \alpha = 1$. This gives $\cos^2 \alpha = \frac{1}{3}$,so $\cos \alpha = \pm \frac{1}{\sqrt{3}}$. Since the angles are given to be acute,$\cos \alpha, \cos \beta, \cos \gamma$ must be positive. Therefore,the direction cosines are $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$.

The direction cosines of a line which makes equal acute angles with the coordinate axes are

Similar Questions

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

The number of straight lines that are equally inclined to the three-dimensional coordinate axes is

Let $O$ be the origin and $P$ be a point which is at a distance of $3$ units from the origin. If the direction ratios of $\vec{OP}$ are $(1, -2, -2)$,then the coordinates of $P$ are

The angle between the lines with direction ratios $\left( \frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2} \right)$ and $\left( \frac{\sqrt{3}}{4}, \frac{1}{4}, -\frac{\sqrt{3}}{2} \right)$ is ......... $^o$.

If $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines,show that the direction cosines of the line perpendicular to both of these are $m_{1} n_{2}-m_{2} n_{1}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-l_{2} m_{1}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module