A line makes angles alpha, beta, gamma with t | vedclass

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

Question

(B) We know that the sum of the squares of the direction cosines of a line is $1$,i.e.,$\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$. Given

Vedclass · Accepted Answer

$90$

Vedclass · Answer

(B) We know that the sum of the squares of the direction cosines of a line is $1$,i.e.,$\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$. Given that $\alpha + \beta = 90^{\circ}$,we have $\alpha = 90^{\circ} - \beta$. Substituting this into the expression for $\cos \alpha$,we get $\cos \alpha = \cos(90^{\circ} - \beta) = \sin \beta$. Therefore,$\cos^{2} \alpha = \sin^{2} \beta$. Using the identity $\sin^{2} \beta = 1 - \cos^{2} \beta$,we get $\cos^{2} \alpha = 1 - \cos^{2} \beta$,which implies $\cos^{2} \alpha + \cos^{2} \beta = 1$. Substituting this into the original equation: $(1) + \cos^{2} \gamma = 1$. This simplifies to $\cos^{2} \gamma = 0$,which means $\cos \gamma = 0$. Thus,$\gamma = 90^{\circ}$.

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

Similar Questions

The direction cosines of the line passing through $P(2, 3, -1)$ and the origin $O(0, 0, 0)$ are:

The direction cosines $l, m, n$ of two lines satisfy the equations $3l + m + 5n = 0$ and $6mn - 2nl + 5lm = 0$. If $\theta$ is the angle between these lines, then $|\cos \theta| = $

The direction cosines of three lines passing through the origin are $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$,and $(l_3, m_3, n_3)$. The lines will be coplanar if

If $\theta$ is the acute angle between the two lines whose direction cosines are connected by the relations $l+m+n=0$ and $2lm+2nl-mn=0$, then $\cos \theta=$

If the direction cosines of two lines are such that $2l + m + 2n = 0$ and $3l^2 + 5m^2 - 11n^2 = 0$,then the angle between the two lines is

Vedclass Test Series

Exam Paper Generator

Online Exam Module