Problem 27
Question
The angle between the straight lines whose direction cosines are given by \(2 l+2 m-n=0, m n+n l+l m=0\), is (A) \(\frac{\pi}{2}\) (B) \(\frac{\pi}{3}\) (C) \(\frac{\pi}{=}\) (D) none of these
Step-by-Step Solution
Verified Answer
The angle is \(\frac{\pi}{2}\), option A.
1Step 1: Identify the given equations
We have two equations involving the direction cosines of two lines: 1. \(2l + 2m - n = 0\) 2. \(mn + nl + lm = 0\) These equations describe relations between the direction cosines \(l, m,\) and \(n\).
2Step 2: Recognize conditions for direction cosines
Direction cosines \((l, m, n)\) must satisfy \(l^2 + m^2 + n^2 = 1\) due to the unit vector condition. Lines corresponding to different vectors should satisfy these geometric conditions in 3D space.
3Step 3: Apply elimination method
To solve for \(l, m,\) and \(n\), we need to subtract and manipulate these equations to isolate and express these variables in terms of each other. This typically starts with expressing one variable linearly in terms of others from one equation.
4Step 4: Solve for a relationship among variables
From the first equation \(2l + 2m = n\), substitute \(n\) in the second equation to find a relatable expression among \(l\) and \(m\). This later aids in solving for individual values of the cosines.
5Step 5: Translate to angle using cosine theorem formula
Utilize the dot product relationship between directions of the lines: \(\cos\theta = l_1l_2 + m_1m_2 + n_1n_2\). Once you have identified \((l, m, n)\) for both lines, use this cosine formula to evaluate \(\theta\), which is the angle between them.
Key Concepts
Understanding Direction CosinesExploring the Equations of LinesUsing Trigonometric Relationships for Angles
Understanding Direction Cosines
Direction cosines play a key role in understanding the orientation of a line in three-dimensional space. They are essentially the cosines of angles that a line makes with each of the coordinate axes. If you imagine a line coming out of a 3D graph and touching each axis, the direction cosines tell you the angle of that interaction.
Direction cosines \((l, m, n)\) are part of a unit vector, meaning they must satisfy the equation \(l^2 + m^2 + n^2 = 1\). This equation ensures that the vector representing the line has a length, or magnitude, of 1. It's like ensuring the line isn't "stretched", so to speak. For different lines, these cosines wield the power to define multiple possible orientations in the 3D space.
Understanding and manipulating these direction cosines are crucial when calculating the angles between lines, as this mathematical process allows us to express the geometry of the lines in mathematical terms that are easier to work with.
Direction cosines \((l, m, n)\) are part of a unit vector, meaning they must satisfy the equation \(l^2 + m^2 + n^2 = 1\). This equation ensures that the vector representing the line has a length, or magnitude, of 1. It's like ensuring the line isn't "stretched", so to speak. For different lines, these cosines wield the power to define multiple possible orientations in the 3D space.
Understanding and manipulating these direction cosines are crucial when calculating the angles between lines, as this mathematical process allows us to express the geometry of the lines in mathematical terms that are easier to work with.
Exploring the Equations of Lines
The equations that describe the properties of lines can seem a bit complex at first, but they simply encode all the information regarding the geometry of a line. Specifically, when we talk about the equations of lines in terms of direction cosines, they allow us to make geometric sense of a line's orientation.
In this context, equations like \(2l + 2m - n = 0\) and \(mn + nl + lm = 0\) reveal how the direction cosines are interrelated for the lines in question. By satisfying these equations, the direction cosines must adjust to maintain the specific geometric relations between the lines.
Each of these equations acts as a condition or rule that the cosines must fulfill, almost like a mathematical puzzle. Solving these equations often involves algebraically manipulating terms to isolate variables or express them in relation to one another. This stepwise breaking down into simpler terms helps untangle the underlying geometry they encode.
In this context, equations like \(2l + 2m - n = 0\) and \(mn + nl + lm = 0\) reveal how the direction cosines are interrelated for the lines in question. By satisfying these equations, the direction cosines must adjust to maintain the specific geometric relations between the lines.
Each of these equations acts as a condition or rule that the cosines must fulfill, almost like a mathematical puzzle. Solving these equations often involves algebraically manipulating terms to isolate variables or express them in relation to one another. This stepwise breaking down into simpler terms helps untangle the underlying geometry they encode.
Using Trigonometric Relationships for Angles
When calculating the angle between two lines, trigonometry becomes our tool of choice. Specifically, the cosine of the angle between the lines can be computed using their direction cosines through the dot product formula.
This relationship can be expressed as: \: \[\cos\theta = l_1l_2 + m_1m_2 + n_1n_2\]\ where \(l_1, m_1, n_1\) and \(l_2, m_2, n_2\) are the direction cosines of the two lines, respectively. This formula effectively allows the transition from the algebra of direction cosines to the geometry of angles.
The dot product \(l_1l_2 + m_1m_2 + n_1n_2\) yields a single scalar that can be directly interpreted as the cosine of the angle between the lines. From there, good old trigonometry takes over to find that angle \(\theta\), providing a clear-cut measure of how the lines relate spatially.
This relationship can be expressed as: \: \[\cos\theta = l_1l_2 + m_1m_2 + n_1n_2\]\ where \(l_1, m_1, n_1\) and \(l_2, m_2, n_2\) are the direction cosines of the two lines, respectively. This formula effectively allows the transition from the algebra of direction cosines to the geometry of angles.
The dot product \(l_1l_2 + m_1m_2 + n_1n_2\) yields a single scalar that can be directly interpreted as the cosine of the angle between the lines. From there, good old trigonometry takes over to find that angle \(\theta\), providing a clear-cut measure of how the lines relate spatially.
Other exercises in this chapter
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