If the direction cosines of two lines are suc | vedclass

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

Question

(C) Given equations for direction cosines $(l, m, n)$ are: $l+m+n=0$  --- $(i)$ $l^2+m^2-n^2=0$ --- $(ii)$ From $(i)$, $n = -(l+m)$. Substituting this

Vedclass · Accepted Answer

$\frac{\pi}{3}$

Vedclass · Answer

(C) Given equations for direction cosines $(l, m, n)$ are: $l+m+n=0$  --- $(i)$ $l^2+m^2-n^2=0$ --- $(ii)$ From $(i)$, $n = -(l+m)$. Substituting this into $(ii)$: $l^2 + m^2 - (-(l+m))^2 = 0$ $l^2 + m^2 - (l^2 + m^2 + 2lm) = 0$ $-2lm = 0 \Rightarrow lm = 0$. This implies either $l=0$ or $m=0$. Case $1$: If $l=0$, then from $(i)$, $m+n=0 \Rightarrow m=-n$. Let $m=1$, then $n=-1$. The direction ratios are $(0, 1, -1)$. Case $2$: If $m=0$, then from $(i)$, $l+n=0 \Rightarrow l=-n$. Let $l=1$, then $n=-1$. The direction ratios are $(1, 0, -1)$. Let the two vectors be $\vec{a} = 0\hat{i} + 1\hat{j} - 1\hat{k}$ and $\vec{b} = 1\hat{i} + 0\hat{j} - 1\hat{k}$. The angle $	heta$ between them is given by $\cos 	heta = \frac{|\vec{a} \cdot \vec{b}|}{|\vec{a}| |\vec{b}|}$. $\vec{a} \cdot \vec{b} = (0)(1) + (1)(0) + (-1)(-1) = 1$. $|\vec{a}| = \sqrt{0^2 + 1^2 + (-1)^2} = \sqrt{2}$. $|\vec{b}| = \sqrt{1^2 + 0^2 + (-1)^2} = \sqrt{2}$. $\cos 	heta = \frac{1}{\sqrt{2} \cdot \sqrt{2}} = \frac{1}{2}$. Therefore, $	heta = \frac{\pi}{3}$.

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

Similar Questions

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 $< a, b, c >$ and $< a', b', c' >$ are the direction ratios of two perpendicular lines,then which of the following is true?

If the angle between lines $AB$ and $CD$ is $\theta$,then the projection of line segment $AB$ on $CD$ is = ..........

The direction ratios of the diagonal of a cube that joins the origin to the opposite corner are (when the $3$ concurrent edges of the cube are the coordinate axes):

If a line makes angles of $120^{\circ}$ and $60^{\circ}$ with the $x$ and $y$ axes respectively,what angle does it make with the $z$ axis?

Vedclass Test Series

Exam Paper Generator

Online Exam Module