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------------------------------------------------ If [tex]l_1, m_1, n_1[/tex] and [tex]l_2, m_2, n_2[/tex] are the direction cosines of two vectors and [tex]\theta[/tex] is the angle between them, then the value of [tex]\cos \theta[/tex] is:

A. [tex]l_1 l_2 + m_1 m_2 + n_1 n_2[/tex]

B. [tex]l_1 m_1 + m_1 n_1 + n_1 l_1[/tex]

C. [tex]l_2 m_2 + m_2 n_2 + n_2 l_2[/tex]

D. [tex]m_1 l_2 + l_2 m_2 + n_1 m_2[/tex]

The cosine of the angle between two vectors with direction cosines l1, m1, n1 and l2, m2, n2 is given by the dot product formula, which is cosθ = l1l2 + m1m2 + n1n2, corresponding to option A.

Explanation:

The cosine of the angle between two vectors using their direction cosines, one essential formula to remember is cosθ = l1l2 + m1m2 + n1n2. The equation represents the dot product of the two unit vectors, which can be expressed in terms of their direction cosines. In essence, the dot product of two vectors A and B is given by A · B = |A||B|cosθ, which when A and B are unit vectors (having magnitudes of 1), simplifies to just the sum of the products of their corresponding direction cosines. Therefore, the correct answer is option A: l1l2 + m1m2 + n1n2.