If a line L makes angles pi / 3 and pi / 4 wi | vedclass

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

Question

(D) Let the direction cosines of line $L$ be $(l, m, n)$.Given that $L$ makes angles $\pi / 3$ and $\pi / 4$ with $Y$-axis and $Z$-axis respectively.T

Vedclass · Accepted Answer

$\cos^{-1} \left(\frac{\sqrt{2}+1}{\sqrt{6}}\right)$

Vedclass · Answer

(D) Let the direction cosines of line $L$ be $(l, m, n)$.Given that $L$ makes angles $\pi / 3$ and $\pi / 4$ with $Y$-axis and $Z$-axis respectively.Therefore,$m = \cos(\pi / 3) = 1 / 2$ and $n = \cos(\pi / 4) = 1 / \sqrt{2}$.We know that $l^2 + m^2 + n^2 = 1$.Substituting the values,$l^2 + (1 / 2)^2 + (1 / \sqrt{2})^2 = 1 \Rightarrow l^2 + 1 / 4 + 1 / 2 = 1 \Rightarrow l^2 = 1 - 3 / 4 = 1 / 4$.Thus,$l = 1 / 2$ (taking the positive value).The direction ratios of the second line are $(1, 1, 1)$. Its direction cosines are $(1 / \sqrt{3}, 1 / \sqrt{3}, 1 / \sqrt{3})$.The angle $	heta$ between the two lines is given by $\cos 	heta = |l_1 l_2 + m_1 m_2 + n_1 n_2|$.$\cos 	heta = |(1 / 2)(1 / \sqrt{3}) + (1 / 2)(1 / \sqrt{3}) + (1 / \sqrt{2})(1 / \sqrt{3})| = |1 / (2 \sqrt{3}) + 1 / (2 \sqrt{3}) + 1 / \sqrt{6}|$.$\cos 	heta = |1 / \sqrt{3} + 1 / \sqrt{6}| = |\sqrt{2} / \sqrt{6} + 1 / \sqrt{6}| = (\sqrt{2} + 1) / \sqrt{6}$.Therefore,$	heta = \cos^{-1} \left( \frac{\sqrt{2} + 1}{\sqrt{6}} ight)$.

If a line $L$ makes angles $\pi / 3$ and $\pi / 4$ with $Y$-axis and $Z$-axis respectively,then the angle between $L$ and another line having direction ratios $(1, 1, 1)$ is

Similar Questions

$A$ 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 $\alpha$ made with the $x$-axis is acute.

$A$ line $AB$ in three-dimensional space makes angles $45^{\circ}$ and $120^{\circ}$ with the positive $x$-axis and the positive $y$-axis respectively. If $AB$ makes an acute angle $\theta$ with the positive $z$-axis,then $\theta$ equals

If a line makes angles $\alpha, \beta, \gamma$ with the three-dimensional coordinate axes respectively,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = $

$A$ straight line is equally inclined to all the three coordinate axes. Then,the angle made by the line with the $y$-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