A line makes the same angle theta with each o | vedclass

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

Question

(C) Let the direction cosines of the line be $(\cos 	heta, \cos \beta, \cos 	heta)$.Since the sum of the squares of the direction cosines is $1$,we

Vedclass · Accepted Answer

$\frac{3}{5}$

Vedclass · Answer

(C) Let the direction cosines of the line be $(\cos 	heta, \cos \beta, \cos 	heta)$.Since the sum of the squares of the direction cosines is $1$,we have:$\cos^2 	heta + \cos^2 \beta + \cos^2 	heta = 1$$2\cos^2 	heta + \cos^2 \beta = 1$Using the identity $\cos^2 \alpha = 1 - \sin^2 \alpha$,we get:$2(1 - \sin^2 	heta) + (1 - \sin^2 \beta) = 1$$3 - 2\sin^2 	heta - \sin^2 \beta = 1$Given $\sin^2 \beta = 3\sin^2 	heta$,substitute this into the equation:$3 - 2\sin^2 	heta - 3\sin^2 	heta = 1$$3 - 5\sin^2 	heta = 1$$5\sin^2 	heta = 2$$\sin^2 	heta = \frac{2}{5}$Now,calculate $\cos^2 	heta$:$\cos^2 	heta = 1 - \sin^2 	heta = 1 - \frac{2}{5} = \frac{3}{5}$

$A$ line makes the same angle $\theta$ with each of the $x$ and $z$ axes. If the angle $\beta$ which it makes with the $y$-axis is such that $\sin^2 \beta = 3\sin^2 \theta$,then $\cos^2 \theta$ is-

Similar Questions

If the direction cosines of two lines are given by $l+3m+5n=0$ and $5lm-2mn+6ln=0$,then the angle between the lines is

Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+m-n=0$ and $l^{2}+m^{2}-n^{2}=0$. Then the value of $\sin^{4} \alpha + \cos^{4} \alpha$ is

$ABC$ is a triangle in a plane with vertices $A(2, 3, 5)$,$B(-1, 3, 2)$,and $C(\lambda, 5, \mu)$. If the median through $A$ is equally inclined to the coordinate axes,then the value of $(\lambda^3 + \mu^3 + 5)$ is

If $\left( \frac{1}{2}, \frac{1}{3}, n \right)$ are the direction cosines of a line,then the value of $n$ is

The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$ and $(l_3, m_3, n_3)$ are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module