The direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes. For line $L_1$, these are $(l_1, m_1, n_1)$ and for line $L_2$, these are $(l_2, m_2, n_2)$.
We can represent these lines as unit vectors in 3D space: $$ \vec{u} = l_1\hat{i} + m_1\hat{j} + n_1\hat{k} $$ $$ \vec{v} = l_2\hat{i} + m_2\hat{j} + n_2\hat{k} $$
The angle $\theta$ between two vectors is given by the dot product formula: $$ \cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} $$ Since $\vec{u}$ and $\vec{v}$ are unit vectors, their magnitudes are 1. Thus: $$ \cos\theta = (l_1)(l_2) + (m_1)(m_2) + (n_1)(n_2) $$
The question specifies $\theta$ as the acute angle. The dot product $l_1l_2 + m_1m_2 + n_1n_2$ can result in a negative value if the angle is obtuse. To ensure the result represents the acute angle, we take the absolute value: $$ \cos\theta = |l_1l_2 + m_1m_2 + n_1n_2| $$
Final Answer: D
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