Question

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^{2}+m^{2}+n^{2}$ is (a) $\frac{\pi}{6}$ (b) $\frac{\pi}{2}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{4}$

Name: Problem 6, Chapter 26: 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics
Uploaded: 2021-11-05T11:07:36-08:00
Duration: 3 min 21 s
Channel: Sriparna Bhattacharjee
Description: The angle between the lines whose direction cosines satisfy the equations l+m+n=0 and l^2+m^2+n^2 is (a) (π)/(6) (b) (π)/(2) (c) (π)/(3) (d) (π)/(4)

   The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^{2}+m^{2}+n^{2}$ is 
(a) $\frac{\pi}{6}$
(b) $\frac{\pi}{2}$
(c) $\frac{\pi}{3}$
(d) $\frac{\pi}{4}$

43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics

Disha Experts 16th Edition

Chapter 26, Problem 6

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Solved by Expert Sriparna Bhattacharjee

11/05/2021

Step 1

Step 1: The given equations are $l+m+n=0$ and $l^{2}+m^{2}+n^{2}=1$. Show more…

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The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^{2}+m^{2}+n^{2}$ is (a) $\frac{\pi}{6}$ (b) $\frac{\pi}{2}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{4}$

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Key Concepts

Direction Cosines

Direction cosines are the cosines of the angles that a line makes with the x, y, and z axes. They serve as components of a unit vector that represents the orientation of the line in three-dimensional space, and hence they satisfy the normalization condition l² + m² + n² = 1.

Linear Constraint on Direction Cosines

A linear equation involving direction cosines, like l + m + n = 0, acts as a constraint that restricts the possible orientations of the line. This reduces the degrees of freedom of the direction cosines by confining them to a specific plane within the space of all possible directions.

Angle Between Lines

The angle between two lines in space is calculated using the dot product of their direction vectors. When the direction cosines of the lines are subject to a constraint, the dot product approach can sometimes yield a constant angle between any two lines from the family, independent of the specific choice of direction cosines.

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