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) \(\frac{\pi}{4}\) (D) none of these
Step-by-Step Solution
Verified Answer
The angle \( \theta \) is \( \frac{\pi}{2} \) (Option A).
1Step 1: Understand the given equations
Two equations involving direction cosines of straight lines are provided: \( 2l + 2m - n = 0 \) and \( mn + nl + lm = 0 \). These equations are linear and quadratic homogeneous equations respectively, which relate to the directional cosines \( l, m, \) and \( n \).
2Step 2: Identify correlation to trigonometric identity
Recall the identity for angle between lines in terms of direction cosines. If \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) are direction cosines for two lines, the cosine of the angle \( \theta \) between them can be given by \( \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \). In this case, use the identities to solve for respective direction cosines.
3Step 3: Solve the simultaneous equations
Use the first equation to express one variable in terms of the others, say \( n = 2l + 2m \). Substitute into the second equation \( mn + nl + lm = 0 \). This yields \( m(2l + 2m) + (2l + 2m)l + lm = 0 \). Simplify and solve these for relationships between \( l, m, \) and \( n \).
4Step 4: Calculate the angle using direction cosine identity
From step 3, calculate direction cosines individually. Substitute them into the identity \( \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \) to find \( \cos \theta \). Evaluate \( \theta \) using trigonometric functions to match with given options.
5Step 5: Calculate \( \theta \) and Compare
Substitute the solved direction cosines into the cosine of angle formula and calculate \( \theta \). In this problem, ensure the angle derived matches one of the given options, considering their respective radians.
Key Concepts
Direction CosinesLinear EquationsTrigonometric IdentitySimultaneous Equations
Direction Cosines
Direction cosines are essential in 3D geometry as they provide a way to describe the orientation of a line in space. These are the cosines of the angles that a line makes with the coordinate axes. For a line with direction cosines \( l, m, \) and \( n \), this means that the line forms angles \( \alpha \), \( \beta \), and \( \gamma \) with the x, y, and z-axes respectively.
\( l = \cos \alpha \)
\( m = \cos \beta \)
\( n = \cos \gamma \)
These cosines satisfy the fundamental relation \( l^2 + m^2 + n^2 = 1 \) because they are projections on the unit sphere. This property is derived from the Pythagorean theorem as the total length is preserved across these axes. Understanding direction cosines help us find the angle between lines by exploiting the relationships between their direction cosines.
\( l = \cos \alpha \)
\( m = \cos \beta \)
\( n = \cos \gamma \)
These cosines satisfy the fundamental relation \( l^2 + m^2 + n^2 = 1 \) because they are projections on the unit sphere. This property is derived from the Pythagorean theorem as the total length is preserved across these axes. Understanding direction cosines help us find the angle between lines by exploiting the relationships between their direction cosines.
Linear Equations
Linear equations represent straight lines in algebra. In our exercise, the equation \( 2l + 2m - n = 0 \) illustrates such a linear equation involving direction cosines. Solving this type of equation typically involves manipulating variables to express one in terms of others. Here, by rearranging terms, we can set \( n = 2l + 2m \).
This step is crucial as it simplifies the approach to solving for the unknowns. Linear equations like these are vital tools for modeling relationships and can be extended to various applications in geometry and physics. Understanding how to solve and manipulate these equations is key to tackling complex problems in mathematics.
This step is crucial as it simplifies the approach to solving for the unknowns. Linear equations like these are vital tools for modeling relationships and can be extended to various applications in geometry and physics. Understanding how to solve and manipulate these equations is key to tackling complex problems in mathematics.
Trigonometric Identity
Trigonometric identities help simplify the analysis of angles between lines. In case of lines described by direction cosines, the cosine of the angle \( \theta \) between two lines given by direction cosines \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \) is determined using the formula:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]
This identity allows us to connect algebraic properties of lines to their geometric interpretations. By expressing \( \cos \theta \) directly, one can reverse-engineer the relationship to find the precise angle if the direction cosines are known. This trigonometric tool is fundamental in calculations involving angles in various dimensions.
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]
This identity allows us to connect algebraic properties of lines to their geometric interpretations. By expressing \( \cos \theta \) directly, one can reverse-engineer the relationship to find the precise angle if the direction cosines are known. This trigonometric tool is fundamental in calculations involving angles in various dimensions.
Simultaneous Equations
Simultaneous equations are a set of equations with multiple variables that are solved together to find a common solution. In this problem, we encounter two equations: one linear and one quadratic \( 2l + 2m - n = 0 \) and \( mn + nl + lm = 0 \). Solving them involves substitution or elimination methods.
In this scenario, substituting \( n = 2l + 2m \) into the quadratic produces another equation in terms of \( l \) and \( m \). Solving these simultaneous equations gives us potential values for direction cosines that satisfy both conditions. Mastery of this technique is advantageous in many fields, as it provides a systematic approach to finding solutions to complex systems.
In this scenario, substituting \( n = 2l + 2m \) into the quadratic produces another equation in terms of \( l \) and \( m \). Solving these simultaneous equations gives us potential values for direction cosines that satisfy both conditions. Mastery of this technique is advantageous in many fields, as it provides a systematic approach to finding solutions to complex systems.
Other exercises in this chapter
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