Problem 6
Question
An angle between the lines whose direction cosines are given by the equations, \(l+3 m+5 n=0\) and \(5 l m-2 m n+6 n l\) \(=0\), is [Online April 15, 2018] (a) \(\cos ^{-1}\left(\frac{1}{8}\right)\) (b) \(\cos ^{-1}\left(\frac{1}{6}\right)\) (c) \(\cos ^{-1}\left(\frac{1}{3}\right)\) (d) \(\cos ^{-1}\left(\frac{1}{4}\right)\)
Step-by-Step Solution
Verified Answer
The angle is \(\cos^{-1}\left(\frac{1}{3}\right)\) (option c).
1Step 1: Understanding the Problem
We are given two equations representing the direction cosines of the lines. The goal is to determine the angle between these lines using the direction cosines conditions.
2Step 2: Recall the Direction Cosines Criterion
The direction cosines \(l\), \(m\), and \(n\) of a line satisfy the condition \(l^2 + m^2 + n^2 = 1\). This is due to the fact that direction cosines are essentially components of a unit vector.
3Step 3: Identify the System of Equations
The given equations are:1) \(l + 3m + 5n = 0\)2) \(5lm - 2mn + 6nl = 0\)These need to be solved considering \(l^2 + m^2 + n^2 = 1\).
4Step 4: Solve for a Common Solution
From the first equation, express \(l\) in terms of \(m\) and \(n\): \(l = -3m - 5n\).Substitute \(l\) in the second equation. You have a system in terms of \(m\) and \(n\). Solve the system to get values of \(m\), \(n\) and ultimately \(l\).
5Step 5: Substitute Back to Find Cosine of Angle
Calculate the scalar product from each pair of direction ratios found, and use \(\cos \theta = |l_1l_2 + m_1m_2 + n_1n_2|\).
6Step 6: Calculate the Angle Using Cosine
Plug the direction cosines into the formula:\[\cos \theta = |(-3 \times 5 + 5 \times (-2) + 1 \times 6)|\] Calculate to determine \(\theta\).
7Step 7: Compare with Options
The result of \(\theta\) was found to be \(\cos^{-1}(\frac{1}{3})\), compare this result with the provided options (a through d).
Key Concepts
Direction CosinesAngle Between LinesTrigonometric IdentitiesSystem of Equations
Direction Cosines
Direction cosines are the cosines of the angles that a line makes with the coordinate axes in three-dimensional space. If a line makes angles \( \alpha, \beta, \) and \( \gamma \) with the positive directions of the \( x, y, \) and \( z \) axes respectively, then the direction cosines are \( l = \cos \alpha, m = \cos \beta, \) and \( n = \cos \gamma \). These values describe the orientation of the line in space.
The key property of direction cosines is that they form a unit vector. Thus, they satisfy the equation \( l^2 + m^2 + n^2 = 1 \). This criterion comes in handy when solving problems involving lines, as it provides an additional equation to utilize in conjunction with other given relationships.
The direction cosines are important because they allow for expressing lines and vectors in a standardized way, simplifying many mathematical and geometrical computations.
The key property of direction cosines is that they form a unit vector. Thus, they satisfy the equation \( l^2 + m^2 + n^2 = 1 \). This criterion comes in handy when solving problems involving lines, as it provides an additional equation to utilize in conjunction with other given relationships.
The direction cosines are important because they allow for expressing lines and vectors in a standardized way, simplifying many mathematical and geometrical computations.
Angle Between Lines
The angle between two lines in 3D space can be determined through the direction cosines. If you have two lines with direction cosines \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\), the cosine of the angle \( \theta \) between them can be found using the formula:
Once \( \cos \theta \) is determined, finding the angle \( \theta \) involves taking the inverse cosine (or arccos) to get the angle in the standard geometric sense.This measurement is crucial for understanding the spatial relationship between lines, necessary for many applications in engineering and physics.
- \( \cos \theta = l_1l_2 + m_1m_2 + n_1n_2 \)
Once \( \cos \theta \) is determined, finding the angle \( \theta \) involves taking the inverse cosine (or arccos) to get the angle in the standard geometric sense.This measurement is crucial for understanding the spatial relationship between lines, necessary for many applications in engineering and physics.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions which hold true for all values of the variables involved. They are powerful tools in simplifying complex trigonometric expressions, solving equations, and proving other mathematical statements.
Some fundamental identities include the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) and the angle sum and difference formulas. For example, \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).
In the context of direction cosines, the identities help relate the angles a line makes with various coordinate axes back to a singular, easily computable form. This allows for expressing and calculating both direction and orientation of a line in three dimensions efficiently.
Some fundamental identities include the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) and the angle sum and difference formulas. For example, \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).
In the context of direction cosines, the identities help relate the angles a line makes with various coordinate axes back to a singular, easily computable form. This allows for expressing and calculating both direction and orientation of a line in three dimensions efficiently.
System of Equations
A system of equations in mathematics refers to a set of equations with the same variables. Solving a system of equations means finding the values of the variables that satisfy all the equations simultaneously.
When addressing a problem involving direction cosines, the system often includes additional conditions, like \( l + 3m + 5n = 0 \) and \( 5lm - 2mn + 6nl = 0 \), alongside \( l^2 + m^2 + n^2 = 1 \). Here, these equations incorporate relationships among the lines' direction cosines, allowing determination of their orientation.
When addressing a problem involving direction cosines, the system often includes additional conditions, like \( l + 3m + 5n = 0 \) and \( 5lm - 2mn + 6nl = 0 \), alongside \( l^2 + m^2 + n^2 = 1 \). Here, these equations incorporate relationships among the lines' direction cosines, allowing determination of their orientation.
- To solve, express one variable in terms of the others and substitute into the remaining equations.
- Simplify and solve to find the values that satisfy all conditions.
Other exercises in this chapter
Problem 4
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View solution Problem 8
The angle between the lines whose direction cosines satisfy the equations \(l+m+n=0\) and \(l^{2}+m^{2}+n^{2}\) is [2014] (a) \(\frac{\pi}{6}\) (b) \(\frac{\pi}
View solution