If `l_(1),m_(1),n_(1)` and `l_(2),m_(2),n_(2)` are the directional cosines of two vectors and `theta` is the angle between them, then their value of `cos theta` is

Options

A. `l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)`
B. `l_(1)m_(1)+m_(1)n_(1)+n_(1)l_(1)`
C. `l_(2)m_(2)+m_(2)n_(2)+n_(2)l_(2)`
D. `m_(1)l_(2)+l_(2)m_(2)+n_(1)m_(2)`

Explanation

To find the value of \( \cos \theta \) between two vectors given their directional cosines, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Directional Cosines**:
- The directional cosines of a vector are defined as the cosines of the angles that the vector makes with the coordinate axes. For two vectors, we denote their directional cosines as:
- For vector \( \mathbf{A} \): \( l_1, m_1, n_1 \)
- For vector \( \mathbf{B} \): \( l_2, m_2, n_2 \)

2. **Using the Dot Product**:
- The cosine of the angle \( \theta \) between two vectors can be expressed using the dot product:
\[ \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]

3. **Calculating the Dot Product**:
- The dot product \( \mathbf{A} \cdot \mathbf{B} \) can be calculated using their directional cosines:
\[ \mathbf{A} \cdot \mathbf{B} = l_1 l_2 + m_1 m_2 + n_1 n_2 \]

4. **Calculating the Magnitudes**:
- The magnitudes of the vectors \( \mathbf{A} \) and \( \mathbf{B} \) are given by:
\[ |\mathbf{A}| = \sqrt{l_1^2 + m_1^2 + n_1^2} \]
\[ |\mathbf{B}| = \sqrt{l_2^2 + m_2^2 + n_2^2} \]
- Since \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) are directional cosines, we know that:
\[ l_1^2 + m_1^2 + n_1^2 = 1 \quad \text{and} \quad l_2^2 + m_2^2 + n_2^2 = 1 \]
- Therefore, \( |\mathbf{A}| = 1 \) and \( |\mathbf{B}| = 1 \).

5. **Substituting into the Cosine Formula**:
- Since both magnitudes are 1, we can simplify the formula for \( \cos \theta \):
\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{1 \cdot 1} = l_1 l_2 + m_1 m_2 + n_1 n_2 \]

6. **Final Result**:
- Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]

### Conclusion:
The final answer is:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]