If a line makes an angle of frac{pi}{3} with | vedclass

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

Question

(C) Given that the direction angles are $\alpha = \frac{\pi}{3}$ and $\beta = \frac{\pi}{3}$.Let the angle made by the line with the $Z$-axis be $\gam

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(C) Given that the direction angles are $\alpha = \frac{\pi}{3}$ and $\beta = \frac{\pi}{3}$.Let the angle made by the line with the $Z$-axis be $\gamma$.We know the property of direction cosines: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.Substituting the given values:$\cos^2(\frac{\pi}{3}) + \cos^2(\frac{\pi}{3}) + \cos^2 \gamma = 1$.Since $\cos(\frac{\pi}{3}) = \frac{1}{2}$,we have:$(\frac{1}{2})^2 + (\frac{1}{2})^2 + \cos^2 \gamma = 1$.$\frac{1}{4} + \frac{1}{4} + \cos^2 \gamma = 1$.$\frac{1}{2} + \cos^2 \gamma = 1$.$\cos^2 \gamma = 1 - \frac{1}{2} = \frac{1}{2}$.$\cos \gamma = \pm \frac{1}{\sqrt{2}}$.Since the angle $\gamma$ is acute,$\cos \gamma = \frac{1}{\sqrt{2}}$,which implies $\gamma = \frac{\pi}{4}$.

If a line makes an angle of $\frac{\pi}{3}$ with each $X$ and $Y$ axis,then the acute angle made by $Z$-axis is

Similar Questions

If $\theta$ is the angle between the lines whose direction cosines $(l, m, n)$ satisfy the equations $6mn - 2nl + 5lm = 0$ and $3l + m + 5n = 0$,then $\sin \theta = $

The angle between the lines with direction ratios $(2, -2, 1)$ and $(1, -2, 2)$ is

If $OA$ is equally inclined to $OX, OY$ and $OZ$ and if $A$ is $\sqrt{3}$ units from the origin,then $A$ is:

If a line makes an angle of $\frac{\pi}{3}$ with each of the $X$ and $Y$-axes,then the acute angle made by the line with the $Z$-axis is:

If the direction cosines of two lines satisfy the equations $l-2m+n=0$ and $lm+10mn-2nl=0$,and $\theta$ is the angle between the lines,then $\cos \theta=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module