Question:
If \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) are direction cosines of lines \( L_1 \) and \( L_2 \) respectively and \( \theta \) is the acute angle between them, then :
If \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) are direction cosines of lines \( L_1 \) and \( L_2 \) respectively and \( \theta \) is the acute angle between them, then :
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Whenever a question specifies an "acute angle" between lines in 3D geometry, always look for the option containing the absolute value modulus sign to ensure the output value is positive.
Updated On: Jul 4, 2026
- \( \cos \theta = l_1l_2 + m_1m_2 + n_1n_2 \)
- \( \sin \theta = l_1l_2 + m_1m_2 + n_1n_2 \)
- \( \tan \theta = \frac{l_1}{l_2} + \frac{m_1}{m_2} + \frac{n_1}{n_2} \)
- \( \cos \theta = |l_1l_2 + m_1m_2 + n_1n_2| \)
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The Correct Option is D
Solution and Explanation
Concept:
The angle between two lines in three-dimensional space can be determined using the direction cosines of the lines. If two lines have direction cosines \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \), the cosine of the angle \( \theta \) between them is given by their dot-product analogue. Since \( \theta \) is explicitly stated to be an acute angle, \( \cos \theta \) must be non-negative, necessitating the absolute value signs.
Step 1: Standard formula definition.
The unit vectors along the lines \( L_1 \) and \( L_2 \) can be written as: \[ \hat{u}_1 = l_1\hat{i} + m_1\hat{j} + n_1\hat{k} \quad \text{and} \quad \hat{u}_2 = l_2\hat{i} + m_2\hat{j} + n_2\hat{k} \] The cosine of the angle \( \theta \) between these vectors is: \[ \cos \theta = \hat{u}_1 \cdot \hat{u}_2 = l_1l_2 + m_1m_2 + n_1n_2 \]
Step 2: Accounting for the acute angle constraint.
Since lines extend infinitely in both directions, they form two supplementary angles. The problem explicitly specifies that \( \theta \) is the acute angle, meaning \( 0 \le \theta \le \frac{\pi}{2} \). In this quadrant, cosine values are always positive. To guarantee a positive result regardless of the orientation choice of direction cosines, we introduce absolute value brackets: \[ \cos \theta = |l_1l_2 + m_1m_2 + n_1n_2| \] This precisely matches option (D).
Step 1: Standard formula definition.
The unit vectors along the lines \( L_1 \) and \( L_2 \) can be written as: \[ \hat{u}_1 = l_1\hat{i} + m_1\hat{j} + n_1\hat{k} \quad \text{and} \quad \hat{u}_2 = l_2\hat{i} + m_2\hat{j} + n_2\hat{k} \] The cosine of the angle \( \theta \) between these vectors is: \[ \cos \theta = \hat{u}_1 \cdot \hat{u}_2 = l_1l_2 + m_1m_2 + n_1n_2 \]
Step 2: Accounting for the acute angle constraint.
Since lines extend infinitely in both directions, they form two supplementary angles. The problem explicitly specifies that \( \theta \) is the acute angle, meaning \( 0 \le \theta \le \frac{\pi}{2} \). In this quadrant, cosine values are always positive. To guarantee a positive result regardless of the orientation choice of direction cosines, we introduce absolute value brackets: \[ \cos \theta = |l_1l_2 + m_1m_2 + n_1n_2| \] This precisely matches option (D).
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