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 :
Show Hint
Always look for the modulus sign in questions involving the "acute angle" between lines or planes.
For direction ratios \((a, b, c)\), the formula is \(\cos \theta = \left| \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}} \right|\).
Updated On: Sep 10, 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:
• Direction cosines represent the unit vector in the direction of a line.
• If unit vectors along lines \(L_1\) and \(L_2\) are \(\hat{u} = l_1\hat{i} + m_1\hat{j} + n_1\hat{k}\) and \(\hat{v} = l_2\hat{i} + m_2\hat{j} + n_2\hat{k}\), the angle \(\theta\) between them is found via the dot product.
• Since the problem specifies \(\theta\) is the acute angle, the absolute value of the cosine must be taken.
Step 1: Apply the dot product formula
The cosine of the angle between two lines with direction cosines \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\) is given by:
\[ \cos \theta = l_1l_2 + m_1m_2 + n_1n_2 \]
Step 2: Account for the "acute" condition
Between two intersecting lines, two angles are formed: \(\theta\) and \(180^\circ - \theta\).
One is acute and the other is obtuse. The acute angle always has a positive cosine value.
Therefore, we use the modulus sign to ensure the result is positive: \[ \cos \theta = |l_1l_2 + m_1m_2 + n_1n_2| \]
• Direction cosines represent the unit vector in the direction of a line.
• If unit vectors along lines \(L_1\) and \(L_2\) are \(\hat{u} = l_1\hat{i} + m_1\hat{j} + n_1\hat{k}\) and \(\hat{v} = l_2\hat{i} + m_2\hat{j} + n_2\hat{k}\), the angle \(\theta\) between them is found via the dot product.
• Since the problem specifies \(\theta\) is the acute angle, the absolute value of the cosine must be taken.
Step 1: Apply the dot product formula
The cosine of the angle between two lines with direction cosines \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\) is given by:
\[ \cos \theta = l_1l_2 + m_1m_2 + n_1n_2 \]
Step 2: Account for the "acute" condition
Between two intersecting lines, two angles are formed: \(\theta\) and \(180^\circ - \theta\).
One is acute and the other is obtuse. The acute angle always has a positive cosine value.
Therefore, we use the modulus sign to ensure the result is positive: \[ \cos \theta = |l_1l_2 + m_1m_2 + n_1n_2| \]
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