The direction cosines l, m, n of two lines sa | vedclass

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

Question

(C) Given equations are $3l + m + 5n = 0$ and $6mn - 2nl + 5lm = 0$. From the first equation,$m = -3l - 5n$. Substituting this into the second equatio

Vedclass · Accepted Answer

$\frac{1}{6}$

Vedclass · Answer

(C) Given equations are $3l + m + 5n = 0$ and $6mn - 2nl + 5lm = 0$. From the first equation,$m = -3l - 5n$. Substituting this into the second equation: $6n(-3l - 5n) - 2nl + 5l(-3l - 5n) = 0$ $-18nl - 30n^2 - 2nl - 15l^2 - 25nl = 0$ $-15l^2 - 45nl - 30n^2 = 0$ Dividing by $-15$: $l^2 + 3nl + 2n^2 = 0$ $(l + n)(l + 2n) = 0$ This gives two cases: Case $1$: $l = -n$. Substituting into $m = -3l - 5n$,we get $m = -3(-n) - 5n = 3n - 5n = -2n$. The direction ratios are $(-n, -2n, n)$,which simplifies to $(1, 2, -1)$. Case $2$: $l = -2n$. Substituting into $m = -3l - 5n$,we get $m = -3(-2n) - 5n = 6n - 5n = n$. The direction ratios are $(-2n, n, n)$,which simplifies to $(-2, 1, 1)$. Let the direction ratios be $\vec{a} = (1, 2, -1)$ and $\vec{b} = (-2, 1, 1)$. The cosine of the angle $	heta$ is given by: $\cos 	heta = \frac{|\vec{a} \cdot \vec{b}|}{|\vec{a}| |\vec{b}|} = \frac{|(1)(-2) + (2)(1) + (-1)(1)|}{\sqrt{1^2 + 2^2 + (-1)^2} \sqrt{(-2)^2 + 1^2 + 1^2}}$ $\cos 	heta = \frac{|-2 + 2 - 1|}{\sqrt{6} \sqrt{6}} = \frac{|-1|}{6} = \frac{1}{6}$.

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| = $

Similar Questions

If $\alpha, \beta, \gamma$ are the direction angles of a vector and $\cos \alpha = \frac{14}{15}$,$\cos \beta = \frac{1}{3}$,then $\cos \gamma = $

The obtuse angle between the lines whose direction ratios are determined by the equations $a+b+c=0$ and $2ab+2ac-bc=0$ is

The direction cosines of the line $6x - 2 = 3y + 1 = 2z - 2$ are:

$A$ line makes angles $\alpha, \beta, \gamma$ with the coordinate axes. If $\alpha + \beta = 90^\circ$,then $\gamma = \dots \dots ^\circ$.

If a line makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the $X$-axis and $Y$-axis respectively,then the angle made by the line with the $Z$-axis is

Vedclass Test Series

Exam Paper Generator

Online Exam Module