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:
Step 1 (Formula / Governing Law & Given Data): [1/3 Mark]
The angle $\theta$ between two lines with direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ is given by the dot product of their unit direction vectors $\hat{u}_1$ and $\hat{u}_2$. The formula is:
$$\cos \theta = |\vec{u}_1 \cdot \vec{u}_2|$$Where $\vec{u}_1 = l_1\hat{i} + m_1\hat{j} + n_1\hat{k}$ and $\vec{u}_2 = l_2\hat{i} + m_2\hat{j} + n_2\hat{k}$.
Step 2 (Step-by-Step Mathematical Working): [1/3 Mark]
Substituting the components into the dot product formula:
$$\cos \theta = |(l_1\hat{i} + m_1\hat{j} + n_1\hat{k}) \cdot (l_2\hat{i} + m_2\hat{j} + n_2\hat{k})|$$By the definition of the scalar product of vectors:
$$\cos \theta = |l_1 l_2 + m_1 m_2 + n_1 n_2|$$Since the question specifies $\theta$ is the acute angle, the absolute value ensures the cosine value remains positive (as $\cos \theta > 0$ for $0 \le \theta < \pi/2$).
Step 3 (Final Concluded Answer): [1/3 Mark]
Comparing the derived result with the given options, the correct option is (D).
$$\cos \theta = | l_1 l_2 + m_1 m_2 + n_1 n_2 |$$💡 EL10 Concept Insight: Think of direction cosines as coordinates of a point on a unit sphere; the dot product measures how much these two lines "point" in the same direction, and the absolute value bars ensure we always get the smaller (acute) angle between them rather than the larger one.